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An optimization model for the development and response of alluvial river channels Millar, Robert Gary 1994

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AN OPTIMIZATION MODEL FOR THE DEVELOPMENT AND RESPONSE OF ALLUVIAL RIVER CHANNELS  By ROBERT GARY MILLAR B.Sc.(Hons), The University of Queensland, 1984 M.A.Sc., The University of British Columbia, 1991  A THESIS SUBMITfED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1994 © ROBERT GARY MILLAR, 1994  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  3  v  vf€r  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  S4q.r  (9  Abstract 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.  11  Only limited verification of the model has been attempted at this stage by testing the theory on  published field data, and through comparisons with previously obtained qualitative relations. Good agreement between the modelled and observed channel geometries was obtained for channels with weakly developed bank vegetation using a simplified version of the optimization model. The effect of the bank vegetation on the channel width was shown to agree closely with empirical regime analyses.  The potential uses of the optimization model are to interpret observed river adjustments, to predict future channel changes in disturbed catchments, and to gain further understanding and insight into the behaviour of alluvial rivers.  Recommendations are made for future verification of the model.  111  Table of Contents Abstract  ii  Table of Contents  iv  List of Appendices  ix  List of Tables  x  List of Figures  xi  List of Symbols  xiv  Acknowledgement  xx  1 INTRODUCTION  1  1.1 INTRODUCTION  1  1.2 EXAMPLES OF ACTIVITIES THAT IMPACT RIVERS  4  1.2.1 Dams and Reservoirs  4  1.2.2 Urbanization and Agricultural Development  5  1.2.3 Logging  6  1.2.4 Other Activities  7  1.3 APPROACHES TO THE PROBLEM  7  1.3.1 Case Studies  7  1.3.2 Qualitative Relations  8  1.3.3 Empirical Regime Equations  9  1.3.3.1 Width  10  1.3.3.2Depth  11  1.3.3.3 Slope  11  1.4.4 Analytical Models  13  1.3.4.1 StaticModels  13  1.3.4,2 Dynamic Models  14  1.4 THESIS OUTLINE  16  iv  2 QUALITATIVE MODEL FORMULATION  .19  2.1 iNTRODUCTION  19  2.1 TEMPORAL AND SPATIAL SCALES  19  2.2.1 Temporal Scales  19  2.2.2 Spatial Scales  20  2.3 DEFINITION OF EQUILIBRIUM  21  2.4 OPTIMAL HYDRAULIC GEOMETRY  22  2.5 OPTIMLZATION MODEL  24  2.5.1 Independent Variables  26  2.5.2 Dependent Variables  27  2.5.3 Objective Function  27  2.5.4 Constraints  31 32  2.6 SUMMARY  36  3 DISCHARGE CONSTRAiNT 3.1 INTRODUCTION  36  3.2 FLOW RESISTANCE  36  3.2.1 Subdivision off into Grain and Form Components  39  3.2.2 Estimation off” for Gravel-Bed Rivers  42  3.2.2.1 Einstein and Barbarossa (1952)  42  3.2.2.2 Parker and Peterson (1980)  44  3.2.2.3 Prestegaard (1983)  45  3.2.3 Variation of f” with Discharge 3.3 BANKFULL DISCHARGE AND OVERBANK FLOW 3.3.1 Recurrence Interval of Bankfull Flow  46 49 52  3.4 SUBDIVISION OFf INTO BED AND BANK COMPONENTS  53  3.5 DISCHARGE CONSTRAINT  57  3.6 SUMMARY  58  4 BEDLOAD CONSTRAINT  68  4.1 INTRODUCTION  68  v  4.2 BOUNDARY SHEAR STRESS DISTRIBUTION  .68  4.2.1. Cross-Channel Momentum Transfer  .69  4.2.2 Analytical Solutions  70  4.2.3 Experimental Methods  71  4.2.3.1 Non-uniform bed and bank roughness  73 75  4.2.4 Gram and Bar Shear 4.3 MODELLING SEDIMENT TRANSPORTING CAPACITY AT  77  BANKFULL 4.3.1 Generalized Equations for Mean Bed and Bank Shear  78  Stresses  80  4.4 BED SURFACE 4.4.1 Influence of the Armour Layer  82  4.5 PARKER SURFACE-BASED BEDLOAI) TRANSPORT RELATION  84  4.5.1 Calculation of Armour Grain Size Distribution  88  4.5.2 Modification of the Parker Surface-Based Relation for Natural Rivers with Variable Discharge  91  4.6 TOTAL BEDLOAD CONSTRAINT  94  4.7 SUMMARY  95  5 BANK STABILITY CONSTRAINT  108  5.1 INTRODUCTION  108  5.2 COHESIVE BANK SEDIMENT  108 108  5.2.1 Mass Failure 5.2.1.1 Bank Stability and Submergence  112  5.2.1 2 Bank Height Constraint  115 115  5.2.2 Fluvial Erosion 5.2.2.1 Bank ShearConstraint 5.3 NONCOHESIVE BANK SEDIMENT  117 117  5.3.1 Mass Failure  118  5.3.2 FluvialErosion  119  vi  .121  5.4 SUMMARY  6 EFFECT OF BANK STABILITY ON CHANNEL GEOMETRY  127 127  6.1 INTRODUCTION 6.2 BANKFULL : FiXED-CHANNEL-SLOPE OPTIMiZATION MODEL  127  6.2.1. Independent Variables  128  6.2.2. Dependent Variables  129  6.2.3. Objective function  129  6.2.4 Constraints  130  6.2.4.1 Continuity  131  6.2.4.2 Bank Stability Constraint  131  6.2.5. Optimization Scheme: Fixed-Channel-Slope 6.3 NONCOHESIVE BANK SEDIMENT  132 135  6.3.1 Low Density Bank Vegetation  138  6.3.2 Effect of bank vegetation  139  6.3.3 Influence of ç is on Channel Geometry 5  143 146  6.4 COHESIVE BANK SEDIMENT 6.4.1 Bank Stability Routine  147  6.4.2 Data From Charlton et al. (1978)  148  6.4.3 Effect of Bank Vegetation  153  6.4.4 Discussion of March etal. (1993)  153  6.5 BANKFULL MODEL:VARIABLE-CHANNEL-SLOPE  157  6.5.1 Noncohesive Bank Sediment  158  6.5.2. Cohesive Bank Sediment  161 164  6.6 SUMMARY 7 FULL MODEL FORMULATION  181  7.1 INTRODUCTION  181  7.2 MODEL FORMULATION  181 182  7.2.1 Independent Variables  VII  .183  7.2.2 Dependent Variables 7.2.3 Objective Function  184  7.2.4 Constraints  184  7.2.4.1 Continuity  184  7.2.4.2 Bedload  185  7.2.4.3 Bank Stability  186 186  7.2.5 Optimization Scheme 7.3VARJABLEFLOWS  187  7.4 ADJUSTMENT OF THE BED SURFACE COMPOSITION  189  Solution Curves  190  7.4.1 So  -  50 r*D  7.4.2 Effect of Sediment Gradation on  550  193  7.5 EFFECT OF SEDIMENT LOAD  193  7.5.1 Valley Slope Constraint  195  7.6 BANKFULL DISCHARGE AS A DEPENDENT VARIABLE  196  7.6.1 Effect of Sediment Load on Bankfull Discharge  199  7.7 APPLICATION OF THE MODEL  200  7.8 HOW AND WHY DO ALLUVIAL CHANNELS OPTIMIZE?  202  7.9 COMPARISON WITH OTHERNUMEBJCAL MODELS  205  7.11 AN EXAMPLE OF RIVER BEHAVIOUR WITHIN AN OPTIMIZATION FRAMEWORK  210  7.11 SUM14’IARY  211  8 CONCLUSIONS AND RECOMMENDATIONS 8.1 SUMMARY  230 230  8.1.1 Inclusion Sank Stability Analysis  231  8.1.2. Modelling Using The Full Flow-Duration Data  233  8.1.3 Adjustment of the Bed Surface  233  8.1.4 Bankftill Discharge as a Dependent Variable  234  8.1.5 Verification of the Optimization Model  235  8.2 RECOMMENDATIONS FOR FUTURE WORK  236  8.2.1 Channel Form or Bar Roughness  236  viii  8.2.2 Surface-Based Transport Relation Based on t’bed  237  8.2.3 Bank Stability  237  8.2.4 Secondary Currents  238  8.2.5 Formulation for Sand-Bed Rivers  239  8.2.6 Model Verification  240  List of Appendices A BANKFULL: FIXED SLOPE MODEL  261  B BANKFULL: VARIABLE SLOPE MODEL  275  C FULL MODEL FORMULATION  291  D DATA FROM HEY AND THORNE (1986) AND ANDREWS (1984)  314  E DATA FROM CHARLTONETAL. (1978)  319  ix  List of Tables Table 5.1. Parameters for 3 cases of bank stability  114  Table 6.1. Ratios between the observed and modelled values of Wand Y  140  Table 6.2. Summary of ratios of observed channel width divided by the unvegetated channel width  142  Table 6.3. Summary of ç values obtained analytically for the data sets of Andrews (1984) and Hey and Thome (1986)  145  Table 6.4. Results from the analysis of data from Charlton et al. (1978)  151  Table 6.5. Ratios between the observed and modelled values of W, Y, and S for variable slope model  159  Table 6.6. Effect of q on reach 13 from Hey and Thome (1986)  160  Table 7.1. Summary of sediment size parameters from Figs 7.10 and 7.11  193  Table 7.2 Comparison of flow parameters for Qbf= 65 3 m / s and  198  Qbf  135 3 m / s  Table D- 1. Data from Hey and Thorne (1986)  315  Table D-2. Data from Andrews (1984)  318  Table E-1. Data from Charlton et al. (1978)  320  x  List of Figures Figure 1.1 Idealised fluvial system. 18 Figure 2.1 Definition sketch for a representative channel length  33  Figure 2.2 A schematic representation of the optimal geometry in fluvial systems  34  Figure 2.3 Definition sketch for development of the coefficient of sediment transport efficiency  35  i  Figure 3.1 Variation off with respect to Rh / D 30 for selected gravel-bed rivers  60  Figure 3.2 Variation off” with respect to  ZI*D50  for selected gravel-bed  rivers  61  Figure 3.3 Variation of f” with respect to the components of  Z*D5O  for  selected gravel-bed rivers Figure 3.4 Variation off with respect to  62  Q  63  Figure 3.5 Definition sketch of a prismatic channel  64  Figure 3.6 Experimental velocity contours (from Seffin, 1964)  65  Figure 3.7 Recurrence interval for Qbf (from Williams, 1978)  66  Figure 3.8 Subdivision of cross-section into bed and bank sections  67  Figure 4.1 Simplified velocity distribution in a straight open channel  97  Figure 4.2 Area methods for calculating local boundary shear stress  98  Figure 4.3 Boundary shear stress distribution from Lane (1955b)  99  Figure 4.4 Boundary shear stress variation for variable Sand uniform boundary roughness  100  Figure 4.5 Boundary shear stress variation for constant Sand non-uniform boundary roughness  101  Figure 4.6 Grain shear stress versus total shear stress  x  102  Figure 4.7 Discretized flow-duration cue. 103 Figure 4.8 Bedload transport rates versus  Q for Oak Creek,  Oregon  Figure 4.9 Variation of the function G  104 105  Figure 4.10 Variation of the functions oj, and  with  gO  after  Parker (1990)  106  Figure 4.11 Variation of , 50 with d  Q for Oak Creek,  Oregon  107  Figure 5.1 Bank stability analysis for planar failure surface  123  Figure 5.2 Bank stability analysis for the method of slices  124  Figure 5.3 Stability curves  125  Figure 5.4 Three special cases for bank stability analysis  126  Figure 6.1 Flow chart for bankfull:fixed-channel-slope model  166  Figure 6.2 Variation of z and  with 9  167  bed  168  Figure 6.3 Variation of  i  ed  and 9 with  Figure 6.4 Comparison of modelled and observed values of Wand Y for low densities of bank vegetation  169  Figure 6.5 Comparison of modelled and observed values of Wand Y for all categories of bank vegetation density  170  Figure 6.6 Comparison of modelled and observed values of Wand Y for low densities of bank vegetation  171  Figure 6.7 Comparison of modelled and observed values of Wand Y for the data from Chariton et al (1978)  172  Figure 6.8 The values of r calculated from the data of Chariton et al. (1978) for treed (T) and grassed (G) channel banks Figure 6.9 Bank stability curves from March et a! (1993)  173 174  Figure 6.10(a) and (b) Photographs of stream bank erosion from a small creek flowing across a beach at low tide  175  Figure 6.10(c) and (d) Photographs of stream bank erosion from a small creek flowing across a beach at low tide  XII  176  Figure 6.11 Flow chart for bankfull: variable-channel-slope optimization 177  model  178  Figure 6.12 Variation of ,7 and S with bed Figure 6.13 Variation of the optimal values of selected dependent variables  179  as a function of Qbf  Figure 6.14 Variation of the optimal values of selected dependent variables as a 180  function of Gbf  Figure 7.1 Flow-duration curve showing numerical approximation  213  Figure 7.2 Sediment gradation curve showing numerical approximation  214  Figure 7.3 Flow-chart for full optimization model formulation  216  Figure 7.4 Flow chart for bedload subfunction  216  Figure 7.5 Calculated sediment transport rates as a function of discharge  217  Figure 7.6 Variation of d 501 with discharge  218  Figure 7.7 The calculated ô.,  -  0 solution curve for Oak Creek, Oregon r  ...  Figure 7.8 Armour and subarmour grain size distributions for Oak Creek Figure 7.9 c5  220  curve for data from Dietrich et a! (1989)  -  219  221  Figure 7.10 Sediment gradation curves for a range of Ug values  222  Figure 7.11 S  223  -  50 D  curve for a range of 0 g values  Figure 7.12 Effect of sediment load on channel geometry  224  Figure 7.13 Calculated variation of S with Qbf  226  Figure 7.14 Calculated variation of W with Gb for constant and variable  Qbf.  ..  226  Figure 7.15 Solution curve used to demonstrate channel optimization  227  Figure 7.16 Solution curves used for a comparison of numerical models  228  Figure 7.17 A solution curve used to interpret the behaviour of a meandering river that has been straightened  229  xlii  List of Symbols A  =  Cross-sectional Area  ii A  =  Change in A for one time step  a  =  Empirical Coefficient  b  =  Empirical Coefficient  C  =  Parameter from Eqn (4.11)  C  =  Multiplier to Scale D to k  C  =  Chezy Flow Resistance Coefficient (Eqn 3.16)  c  =  Soil Cohesion  c’  =  Soil Cohesion in Terms of Effective Stresses  D  =  Gram Diameter or Size  D  =  Bed Surface Grain Diameter where x% is Finer  D,g  =  Geometric Mean Grain Diameter of Bed Surface from Eqn (4.35)  D.  =  50 bank D  =  Median Grain Diameter ofNoncohesive Bank Sediment  d  =  Bedload or Sub Pavement Grain Diameter where x% is Finer  g 3 d  =  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)  =  Sediment Transport Efficiency (Eqn 2.2)  =  Volume Fraction ofDj in Bed-Surface Layer.  e  Repesentative Grain Diameter for Intervalj from the Sediment Gradation Curve (Eqn 7.2)  xiv  1 F  =  Parameter from Eqn (6.2)  FD  =  Driving Force for Mass Failure of Cohesive Soil  FR  =  Driving Force for Mass Failure of Cohesive Soil  8 F  =  Factor of Safety which Equals the Ratio FR / FD  f  =  Darcy-Weisbach Friction Factor  =  Friction Factor for Q’  =  Fraction of D in Subsurface Sediment  =  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  =  gb  =  Dry Bedload Transport Rate per metre width in Mass Units  g*  =  Dimensionless Bedload Transport Rate per metre width  g  =  Gravitational Acceleration (Assumed  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 failure  I  =  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 Roughness  G  [1  L  =  1  =  Sediment Transport Capacity as a Rate for Q (Generally in Units of kg/s)  =  9.81 m/s ) 2  Length ofRepresentative Channel Reach (Fig 2.1), or Length ofFailure Surface (Fig 5.1) Basal Length of Soil Unit  xv  m  =  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 Unit  N  =  Cohesive Bank Stability Factor (Eqn 5.2)  P  Channel Wetted Perimeter  1 p  Probability of Flow  Q  Discharge 3 (m / s)  Q’  within Discharge Interval i  Qbf  =  Bankfull Discharge  Qd  =  Dominant or Characteristic Discharge  =  Q  Represetative Discharge for ith Interval from Flow-Duration Curve (Eqn 7.2)  =  Arithmetic Mean Annual Discharge  =  Volumetric Sediment Transport Rate (L 3 / T)  ‘lb  =  Volumetric Sediment Transport Capacity per Unit Channel Width (L / T) 2  Rh  =  Hydraulic Radius (without subscripts bed or bank refers to bankfull value)  S  =  Energy, Water Surface, or Channel Slope (Uniform Flow Assumed)  S  =  Valley Slope  SR  =  Soil Shearing Resistance  4S  =  Change in Volume of Sediment Stored within a Channel Reach (Eqn 2.1)  SF  =  Shear Force  s  =  Specific Gravity of Noncohesive Bed and Bank Sediment (Assumed2.65)  U  =  Mean Channel Velocity (without subscript refers to bankfull value)  (4  =  Mean Shear Velocity  u  =  Vertically Averaged Local Velocity (Eqn 4.3)  (.J7) x  W  =  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.ill value)  Y  =  z  Normal Channel Depth (without subscripts bed or bank refers to bankfull value) Cross-Channel Coordinate (Eqn 4.3)  =  Angle of Failure surface (degrees)  =  Dimensionless Over-bank Flow Depth Unit Weight of Water (Assumed = 9810 N/rn ) 3  y  Unit Weight of Sediment =  Drained Unit Weight of Cohesive Bank Soil  =  Total Unit Weight of Cohesive Bank Soil Bouyant Unit Weight of Cohesive Bank Soil  Yb  Ratio D 50 / d 50 (Eqn 4.25) Ratio D• / Dsg (Eqn 4.34)  4.  =  Coefficient of lateral momentum transfer (Eqn 4.3)  =  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 Wavelength Coefficient of Sediment Transport Efficiency  e  =  Bank Angle (degrees)  =  Maximum Angle for which Banks are Stable (°)  p  =  Density of Water (Assumed  a  =  Pore Water Pressure at Base of Soil Unit (Fig 5.2)  =  Standard Deviation Based on the Sedimentological Phi Scale (Eqn 4.37)  =  ) 3 1000 kg/rn  xvii  asg  =  Geometric Standard Deviation (Eqn 7.4).  CYo  =  Value of o# from Oak Creek  t  =  Mean Boundary Shear Stress (N/rn ) 2  =  Dimensionless Shear Stress for the Median Grain Diameter of the Mean Annual Bedload, or Subsurface Sediment. Dimensionless Shear Stress for the Median Grain Diameter of the Bed Surface. Reference Dimensionless Bed Shear Stress (Eqn 4.31)  trD  =  t*bd  =  t*bank  =  tcrzt  =  Critical Shear Stress for Cohesive Soil.  =  Straining Function (Eqn 4.34)  =  Value of w from Oak Creek  =  Kinematic Viscosity of Water (Assumed  =  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  ‘ 3 D 1 5  =  Intensity of the Grain Shear for Bed-Surface (Eqn 3.10) From Einstein (1950)  r  =  Conversion Constant (Eqn 4.55)  v  Critical Dimensionless Shear Stress for the Median Bank Grain Diameter on Channel Bed. Critical Dimensionless Shear Stress for the Median Bank Grain Diameter on Sloping Channel Bank.  Subscripts bed  =  Denotes Channel Bed Portion  bank  =  Denotes Channel Bank Portion XVIII  =  0.000001 2 m / s)  =  0.03 86  1  =  Denotes i th flow  j  =  Denotesj th grain size fraction  obs  =  Denotes the Observed Value  mod  =  Denotes the Modelled Value  total  =  Denotes Total Value  unveg  =  Denotes the Unvegetated Value  Superscripts u  =  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 andf  xix  Acknowledgement I 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 many hours discussing the ideas that are presented in this thesis, as well as many that are not. In addition the members of my supervisoly committee, Drs Olav Slaymaker, Bill Castleton, and Dennis 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 University Examiners, Drs Michael Church, and Dave McClung, and by the External Examiner Dr. Dale Bray of the University of New Brunswick. They are to be thanked for the diligence with which they pursued their assigned task.  This research has been funded by a University Graduate Fellowship, an Earl Peterson Memorial Scholarship in Civil Engineering, and by the National Sciences and Engineering Research Council of Canada. I am grateful to all of these sources for providing to me the financial  support to complete this work.  Finally I would like to thank my wife Karen for her support throughout these years of graduate studies.  xx  CHAPTER 1 INTRODUCTION  1.1 INTRODUCTION  Rivers represent an integral component of our civilization and economy. Civilization has historically been concentrated along riparian zones and floodplains due to the availability of fresh water and food resources, the presence of level arable land, and because rivers act as transportation corridors for travel or trade, and as receiving waters for domestic and industrial waste.  In a classic paper, Mackin (1948) emphasised that the river is part of drainage system and that it cannot be understood apart from that system. That is changes which are observed in a river channel can only be understood if they are interpreted within the framework of larger basinscale 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 simple model subdivides the fluvial system into 3 zones. Zone 1 is the production or sediment source area where the flows are generated, and the sediment is derived primarily from hillslope processes. Zone 2 is the transfer area where the principal activity is the passage of the flows and the transport of sediment produced in Zone 1. Zone 3 is the deposition zone where  1  sediment is deposited on an alluvial fan, alluvial plain, delta, or in deeper waters. The work in this thesis will deal primarily with channel processes in Zone 2. Recent increases in population and industrialization have resulted in increased development along rivers and throughout the river basin. Water is controlled and regulated to serve a wide variety of purposes including hydroelectric power generation, irrigation, flood control and stormwater engineering, pollution control, navigation improvement, municipal and industrial water supply, recreation, fish and wildlife, and the conservation of soil and water on watershed lands (Linsley et a?., 1992, p. 1-2). These activities can directly impact the river channels, or may alter the basin hydrology and sediment yield.  The work in his thesis will be restricted to single-thread, alluvial gravel-bed rivers. Alluvial rivers are defined as those with bed and banks composed of sediment which is similar to the sediment that is transported by the river. Gravel-bed rivers are by definition rivers with alluvial sediment in the bed of the channel that has a mean diameter greater than 2 mm. Typically most gravel-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 are bedrock controlled, or those that have incised into older sedimentary sequences will not be considered herein. The analysis in this thesis applies to alluvial rivers with mobile beds, that is those rivers which actively transport bed-material sediment, and have the capacity to modilj their channel dimensions.  It is an underlying assumption of equilibrium channel analysis that alluvial rivers develop a mean 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:  2  The fundamental fact of river science, pure and applied, is that (alluvial) channels tend to adjust themselves to average breadths, depths and slopes and meander sizes that depend on (i) the sequence of water discharges imposed on them, (ii) the sequence of sediment discharges acquired by them from the catchment erosion, erosion of their own boundaries, or other sources and (iii) the liability of their cohesive banks to erosion or deposition.  The term equilibrium is generally synonymous with the expression “in regime” which is widely used in engineering circles (Blench, 1957).  Changes in the imposed discharges and sediment load result in adjustments in the river channels and the development of a new hydraulic geometry. The channel adjustments may result in changes that are determined to be undesirable for economic, environmental, or aesthetic reasons 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 hydraulic structures, aggradation of the channel bed which can reduce the channel capacity and result in an increased frequency of over-bank flooding, and changes in the physical nature of the channel that may impact the aquatic habitat.  The basic goal of this thesis is to develop a mathematical model that can be used to calculate the equilibrium hydraulic geometry of alluvial rivers with gravel beds. The potential uses of such a model are to interpret observed river adjustments, to predict future channel changes in disturbed catchments, and to gain further understanding and insight into the behaviour of this class of rivers.  3  1.2 EXAMPLES OF ACTIVITIES THAT IMPACT RIVERS  In this section a selection of land-use activities and their possible impacts on rivers channels will be discussed. The discussion in this section is necessarily general, and is intended to introduce some of the more common effects of selected basin development activities.  1.2.1 Dams and Reservoirs  The construction of dams and reservoirs can have large effects on the downstream channel geometry as a result of changes to the characteristics of the discharge and sediment load. These structures typically regulate the runoff such that the peak flows are reduced, and the flows are more uniform. Furthermore they also act as sediment traps which capture a large proportion, if not 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 the channel downstream, particularly in gravel-bed rivers where a threshold discharge may have to be exceeded before bedload transport commences. Directly downstream of the dam degradation often results because the sediment supply is reduced close to zero, and the river has excess sediment transporting capacity despite the reduction in peak flows. The degradation zone can migrate downstream a considerable distance (Raynov et at., 1986). Bed degradation is particularly common with sand-bed rivers. For gravel-bed rivers the degradation is often reduced considerably by the development of an immobile armour layer.  In other cases aggradation can develop downstream where tributaries deposit sediment into the main channel that no longer has the capacity to transport the sediment. Dams and reservoirs can also result in changes in the channel width. Wolman and Williams (1984) found that the effect of dams on channel width was highly variable from no change, to a 50% reduction, to a 100% increase in channel width.  4  Dams and reservoirs may not affect the total volume of water that is discharged to the downstream channel, only the timing. This is particularly the case for dams constructed primarily for power generation or flood control. However when a dam is used for water supply or irrigation, or there is diversion into or out of the basin, the volume of water, as well as the timing may be affected.  1.2.2 Urbanization and Agricultural Development  Urbanization 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 is concentrated and more efficiently routed along gutters and storm water drains. As a result the storm hydrograph characteristically becomes “flashier”, that is the peak flows are much greater and occur sooner than before urbanization. The low flows are also usually reduced so that previously permanent streams may become ephemeral.  The effect of urbanization on the sediment yield is less clear. Sediment yield may increase during the construction phase, but may ultimately be reduced below natural levels once development is complete (Wolman, 1967). The bank vegetation may also be disturbed during development.  The most characteristic impact to channel morphology following urbanization is an increase in the 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 not fully alluvial.  The conversion of forest to agricultural land may result in increased peak flows and a decrease in the hydrologic response time of the affected area. Although the changes to the runoff response are usually less pronounced than those due to urbanization when compared on a unit area basis, the development of agricultural land typically covers much larger areas than urban 5  areas. Agricultural development may affect a significant proportion of the total catchment and can therefore have a significant effect on the basin hydrology.  The effects of the conversion from forest to agricultural land on catchment hydrology is longterm and when completed the alluvial rivers can be expected to develop a hydraulic geometry that tends to equilibrium with the modified catchment hydrology.  1.2.3 Logging  Certain logging practices, especially clear-cut logging and road construction can significantly influence the watershed runoff and sediment yield characteristics.  Hydrologic changes which may accompany logging are an increase in total runoff and peak flows. A loss of forest cover can reduce interception and evapotranspiration losses. Compacted areas 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 the suspended sediment yield due to surface erosion as a result of soil disturbance during logging and burning activities is well documented (Beschta, 1978). The yield of coarse sediment may also increase due to accelerated mass-wasting processes on logged bill slopes, and due to failures of road and landing fills, especially in steep mountainous areas.  Removal of stream-side vegetation or disturbance of the stream banks can affect the bank stability and result in channel instability. The bank erosion which accompanies the channel destabilization 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 the basin. In medium to large watersheds timber harvesting may be carried out on a “sustainable” 6  basis over an 80 to 100 year rotation. Under these conditions only 1% or so of the catchment area may be affected during any given year, and the net effect on the catchment hydrology may be slight. However for small catchments or individual sub-catchments within the watershed, in any given year logging may affect a significant proportion of the total area, which may have a profound effect on runoff and sediment yield. At the small catchment or sub-catchment scale logging may be viewed as an episodic disturbance with a period of between 20 to 100 years depending upon regeneration rates.  1.2.4 Other Activities  Other 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 a large impact on the channel geometry. The effect of climate change over shorter periods of time are 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 PROBLEM  There are several approaches to the problem of interpreting and predicting channel adjustments due to man-induced or natural causes.  1.3.1 Case Studies  This approach is based upon monitoring and recording observed channel adjustments in order to gain a greater insight into river behaviour. The insight gained from such studies can then be used to assess the future response of the particular river under study, or can be applied to other rivers which possess similar characteristics.  Methods for monitoring river changes include calibrated river sections that are resurveyed periodically over time, or remote sensing data such as sequential air photographs. Another approach is to compare data from a modified catchment with data recorded from nearby 7  control catchments with similar physical characteristics. Longer term adjustments can be studied  from  sedimentological  or  stratigraphic  evidence,  which  may  include  dendrochronological or radiometric dating (Womack and Schumm, 1977).  1.3.2 Qualitative Relations  Qualitative  proportionalities have been  developed  river  by  engineers  and  fluvial  geomorphologists to predict the response of alluvial channels. The best known examples are from Lane (1955a) and Schumm (1969).  Lane (195 5a) suggested that the following relation is useflul when analysing changes in stream morphology:  (1.1)  GbDcxQdS  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 an increase in Gb, the river will tend to restore equilibrium by increasing S for given values of D and Qd.  Schumm (1969) developed this approach fhrther and proposed the following proportionalities:  wY2  (1.2)  Qd  s  W2S  where W = the channel width, Y  =  the mean channel depth,  wavelength. 8  (1.3)  =  sinuosity, and 2= the meander  Relations (1.1) to (1.3) are statements of indefinite proportions and therefore do not give any quantitative information. These relations do however indicate general trends of river adjustments.  1.3.3 Empirical Regime Equations Empirical regime equations have been developed from the measurement and observation of irrigation canals and natural rivers. Much of the early development of these equations was based 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 of three 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 design equations 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 as Leopold 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:  (1.4)  WcsQd  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. More  sophisticated regime equations may include sediment size and load in the regime equations. The equations of Hey and Thorne (1986) will be reviewed here as they were reportedly derived 9  from data from gravel-bed rivers with mobile beds and are therefore most similar in character to the rivers that are modelled in this thesis. The Hey and Thorne study is the first to explicitly include the effect of sediment load on the regime geometry.  1.3.3.1 Width The general equation from Hey and Thorne (1986) for channel width is:  45 W=3.67Q  where W is in metres, and  Qbf =  (1.6)  /s which is assumed to represent 3 the bankfI.ill discharge in m  the dominant channel-forming discharge. The coefficient of determination for Eqn (1.6) is, =0.7884. Hey and Thorne determined that the channel width was independent of sediment 2 r size and load. Bray (1982b) has found that width varied slightly with grain diameter of the bed sediment.  Hey and Thorne determined that the channel width was strongly influenced by the type and density of the bank vegetation, and that Eqn (1.6) could be improved by discriminating the data on the basis of bank vegetation type. The rivers were subdivided into four bank vegetation categories vegetation type I (grassy banks) to vegetation type IV (>50% tree/shrub cover). The effect 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) =O.9577 which represents a significant increase over Eqn (1.6). The bank vegetation is 2 r  10  interpreted as affecting the bank strength and thus its ability to withstand higher shear stresses exerted by the flowing water. This effect will be examined in Chapter 6.  1.3.3.2 Depth The simple hydraulic geometry relation gives:  35 Y=0.33Q  (1.8)  The value of r = 0.8045. 2  =0.8712 with the inclusion of bed material 2 The coefficient of determination is increased to r grain size:  Y= 0.22 Qb 37 D’  where D 50  =  (1.9)  the median bed grain diameter in metres. The mean depth was not significantly  affected by bank vegetation or sediment load.  1.3.3.3 Slope The channel slope equation determined by the Hey and Thorne study is:  3D 9 D G’° ° 0 S = 0.087Q  (1.10)  where D 84 is the grain diameter of the bed sediment for which 84% of the total sediment is finer 0.6285 which indicates considerable unexplained variance. In 2 than. For Equation (1.10) r addition 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 strongly influenced by the sediment load. 11  While the exponents for the width and depth regime equations are similar for canal and river derived 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:  43 SciQ°  (1.12)  Kellerhals (1967) found that the value of the exponent is equal to -0.4, and Bray (1982b) found that it was equal to -0.33. The exponent in Eqn (1.12) is very close to that obtained by the USBR (-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 same exponent (-0.44). Therefore the value of this exponent of around -0.44 appears to be a very significant one for natural rivers.  It is suggested here that the discrepancy between Eqns (1.11) and (1.12) is that for canals the value of S is largely imposed, while alluvial channels can adjust their value of S. Irrigation canals are generally straight and their planform shape is usually affected more by economics and the location of property boundaries and engineering structures, than as a consequence of any process of self-adjustment. Natural rivers however can adjust their value of S, principally through adjustments of sinuosity, such that an equilibrium is established.  The regime equations such as those presented in this section have been of considerable aid in the design of canals or river training activities. However their usefulness for predicting the adjustments to altered hydrologic regime or sediment supply is limited. For example Eqns (1.7) 12  and (1.8) suggest that the values of W and 7 are independent of Gb. Yet Eqn (1.3) from Schumm (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 be somewhat shallower than a narrow channel with resistant banks for the same value of Qbf  1.3.4 Analytical Models  Analytic models are rational, process-based models which solve the governing equations that describe various channel processes such as flow resistance, continuity, momentum, sediment transport and bank stability. These models can be broadly subdivided into static or dynamic models (Hey, 1982).  1.3.4.1 Static Models Static models attempt to model the steady-state, equilibrium hydraulic geometry. The transient period of adjustment is not considered. A solution is obtained by solving the governing equations 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 States Bureau of Reclamation (USBR) which is presented in Lane (1955b). This model combines equations for flow resistance, continuity, the threshold of sediment motion and bank stability to derive solutions for channels where the sediment is at the threshold of movement at every point across the channel perimeter. However an explicit solution is only possible for the narrowest Type B channel, the wider Type A channel requires an additional relation in the form of an empirical regime equation for the channel width. Mobile-bed models require a sediment transport relation in place of the threshold condition.  13  Generally fluvial systems are recognised as being indeterminate (Hey, 1978; 1988) in that there are more unknown variables than there are equations available for solution. The one and only exception 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 these indeterminate systems of equations. Parker (1978) has obtained solutions by using a result obtained from a theoretical analysis whereby the dimensionless shear stress at the center of the channel can only be 20% above critical value for stable, non-eroding banks to develop. Parker’s results apply only to gently curved channels with banks composed of loose, unconsolidated gravel sediment, and cannot be generalised to all gravel rivers.  An additional approach is the formulation of the problem as an optimization model. The governing equations, or constraints, are solved together with an additional condition that some characteristic of the channel is optimised. Examples are Chang (1980) who contends that the total streampower is minimized, and White et a!. (1982) who state that the channel adjusts such that the sediment transport capacity is a maximum.  1.3.4.2 Dynamic Models Dynamic models are used in an attempt to simulate the channel changes with time. The basic governing equations available for solution are momentum, continuity, sediment mass balance which includes a sediment discharge relation, and a bed elevation equation. Models have been  formulated for steady-uniform, and unsteady-non-uniform flow, and using kinematic wave approximations, and full dynamic-wave formulations. The best known model of this type is HEC-6 (HEC, 1974; Thomas and Prasuhn, 1977). The HEC-6 model has undergone several revisions and later versions include provisions for bed armouring.  The governing equations are formulated as partial differential equations which are solved numerically by either finite-difference or finite-element schemes. The initial geometric and 14  hydraulic characteristics of the channel must be specified. Sediment routing is calculated on a size 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 water surface elevation changes. The numerical models have been applied with reasonable success to predicting 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 severe limitation for predicting channel adjustments.  One of the first dynamic models to consider width adjustments is that of Chang (1982). The model is called FLUVIAL-14 and it can be termed a dynamic optimization model. It is similar in form to the basic HEC-6 model, except that width changes are calculated for each time step using the minimum stream power concept. The width changes at each cross-section are such that the stream power is minimised for the total reach length. A principal assumption of this approach is that during the transient channel adjustments the channel is in a state of dynamic equilibrium.  Osman and Thorne (1988) and Thorne and Osman (1988) develop a bank stability analysis for cohesive bank sediment that could be incorporated into existing 1-D dynamic models. Their bank stability analysis considers fluvial erosion and mass failure of cohesive channel banks. The rate of bank erosion and bed aggradation or degradation is calculated for each time step. Mass failure of the bank occurs when the bank exceeds a critical height following bed degradation or over steepening of the banks. The eroded bank material is included in the sediment mass balance.  Further discussion of the various analytical modeling approaches is deferred to Chapter 7 where comparisons with the optimization model developed in this thesis will be undertaken.  15  1.4 THESIS OUTLINE In this thesis a static analytical model will be developed based upon an optimization formulation. This work was commenced in Millar (1991) and is an extension of the models developed by Chang (1980) and White el al. (1982). The formulation in this thesis will be developed specifically for gravel-bed rivers, however the same concepts also apply to sand-bed rivers.  This thesis can be subdivided into two sections Part A and Part B. Part A covers Chapters 2-5 and deals with the theoretical development of the model, and formulation of the objective function and the constraints. Part B includes Chapters 6-8 and covers the computational scheme and the data analysis, together with the final conclusions and recommendations. The content of each chapter is outlined below.  In Chapter 2 the optimization approach to modeling the hydraulic geometry of alluvial rivers is discussed. Concepts of equilibrium and time scales are presented. The objective function is formulated, and the independent and dependent variables are defined.  The development of the discharge constraint is presented in Chapter 3. The discharge constraint includes two components, flow resistance and continuity. The flow resistance relation is developed for gravel-bed rivers. The complete discharge record is utiised and is input into the model 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 the mean bed shear stress to calculate the sediment transport. Empirical relations to estimate the mean bed and bank shear stress values are presented. The Parker (1990) surface-based bedload transport relation is modified for inputs from the flow duration curve.  16  The bank stability constraint is formulated in Chapter 5 for both cohesive and noncohesive bank sediment.  Chapter 6 covers the major computer analysis in the thesis. A simplified optimization model that uses only the bank-full discharge as input is used to investigate the influence of the bank stability on the geometry of channels with both noncohesive and cohesive bank sediment. The work 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 capacity of the channel is calculated using the full range of flows. The model also calculates the grain size distribution of the bed surface. The effect of sediment load on the channel geometry is examined. Potential applications of the model are reviewed, and the optimization model is compared with other numerical approaches.  In Chapter 8 the final conclusions and recommendations are presented including limitations of this model, and proposals for future study.  17  Zone 1 Production  Zone 2 Transfer  Zone 3 Deposition  Figure 1.1. Idealised fluvial system (after Schumm, 1977).  18  CHAPTER 2 QUALITATIVE MODEL FORMULATION  2.1 INTRODUCTION  In this chapter the principal framework of the model will be discussed and the optimization model will be formulated in a qualitative manner. The modeling will treat the river channel as a system that tends to adjust to an equilibrium hydraulic geometry. The fluvial and alluvial channel system will be defined, and the condition for equilibrium discussed. The objective function for the optimization model will be developed.  2.2 TEMPORAL AND SPATIAL SCALES It is necessary when discussing equilibrium in geomorphic systems to define the temporal and spatial scales over which the equilibrium can be considered to operate.  2.2.1 Temporal Scales Cyclic or geologic time (Schumm and Lichty, 1965; Schumm 1977) refers to very long time spans. Over this time scale fundamental changes occur within the river basin, the topography is reduced by erosion, valley slopes are built up, sea levels and climatic conditions can change very dramatically. Graded time (Schumm and Lichty, 1965; Schumm 1977) represents much shorter time intervals. Over a graded time span the topography, valley slopes, sea level, and other fundamental landforms are relatively constant.  The actual time ranges of geologic and graded time scales are not absolute, but depend on the rates of change of the controlling variables such as climate and topography. However in general  19  geologic time might be thought to have a time span in the order of 1,000,000 years, and graded time up to 1,000 years.  Another time scale is referred to as engineering time, which is the typical life of an engineering structure 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, unless otherwise stated, engineering time scales will be assumed.  2.2.2 Spatial Scales The geometry of a river varies along its length. Typically in the downstream direction the discharge increases, the sediment tends to become finer through selective transport and/or abrasion processes, and the bank sediment and vegetation may change. The term representative channel reach will now be introduced (Fig 2.1). The representative channel reach is defined herein 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 mean channel geometry will also be constant over this reach.  The length of this representative channel reach in absolute terms depends upon the rate of downstream change of the channel properties mentioned in the previous paragraph. A gravel-bed river in a mountainous area may experience a rapid downstream increase in discharge, and rapid downstream decrease in the sediment size. In this case the length of the representative channel reach would be relative short, and may be of the order of ten times the channel width. At the other extreme, a large lowlands river, such as the Mississippi River along the mid to lower reaches, may experience relatively constant values of discharge, and sediment load and calibre for extended distances. In the case the representative channel reach may extend for distances possibly up to 500 times the channel width.  20  2.3 DEFINITION OF EQUILIBRIUM  A 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 over an 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 equilibrium geometry restored.  Note that this definition applies to the reach-averaged and not to the local values of the hydraulic geometry. The term steady-state refers to a time-invariant equilibrium. The absolute time period over which this approximation is valid will depend upon the system being studied. It is valid where there are no significant or systematic changes in the mean annual characteristics of the discharge and sediment load, or in the competence of the bank sediment. In general the steadystate 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 sediment continuity equation for a specified time interval:  I--O=ES  where I  =  input of sediment into the up stream end of the channel; 0  (2.1)  =  the output of sediment  from the downstream end of the channel; and I4S = the change in the volume of sediment which is in storage along the reach (Fig 2.1). The value of iS’ can be positive or negative which represents net deposition or net erosion along the channel reach. At equilibrium 0=1 and ES = 0.  21  The definition of equilibrium presented above is consistent with the so called regime theory favoured by engineers that was developed from the observation of stable, non-silting canals constructed in India and Pakistan. A canal was said to be “in regime” Wit was able to contain the flows and transport the sediment load without appreciable deposition or scour. This concept can be 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 the width, depth, and slope of the river) do not show a definite trend over some time interval, and suggested that this may apply over a period of 20 to 40 years.  2.4 OPTIMAL HYDRAULIC GEOMETRY  An 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 dependent variables or degrees of freedom than there are equations for solution. Even for a simplified channel geometry with only 3 degrees of freedom: width, depth, and slope, the solution remains indeterminate 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 in 1914. In flume experiments under conditions of fixed discharge rate and channel slope, it was demonstrated that by varying the width of the flume an optimal value exists where the sediment transporting 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, together with the narrow bed widths over which sediment transport can occur, results in a low total transport rate. Conversely for the large flume widths the depth of flow and the bed shear stress 22  both become small, and hence the total sediment transport rate also becomes small. Between these 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 is shown schematically in Fig 2.2(a). The channel slope and discharge is constant at each point on the solution curve. The numerical procedures for calculating these solution curves will be developed 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 discharge and sediment transporting capacity are now fixed. In this case the optimum is the point where the slope is a minimum. This example is considered to be analogous to most natural alluvial rivers over engineering time scales. The discharge and sediment load are imposed, and the width depth and slope of the channel are dependent variables that develop in response to these imposed values. This is discussed in Section 2.5.  The experimental results of Gilbert (1914) together with the numerical analyses discussed above combine 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 channel will tend to adjust to this optimum. and that this optimum corresponds to the equilibrium hydraulic geometry.  The preceding assumption is referred to in physical sciences as a “variational principle”. The idea behind a variational principle is that a physical system will select the most “economical” path or mode that requires the least “expenditure” (Konopinski, 1969; p. 169). A branch of mathematics called 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, and most scientists who employ variational principles rely on their intuition in the initial formulation 23  (Kyrala, 1967; P. 119). The properties that the solutions must possess (when their existence has been assumed), can be easier to investigate, both theoretically and experimentally, than the functional relation that describes the system. By employing a variational principle the complexity of the problem can be reduced, and it provides a point from which to commence an investigation into the behaviour of a complex system.  2.5 OPTIMIZATION MODEL Optimization models have been developed previously by Chang (1980), Yang et a!. (1981), and White eta!. (1982) among others to predict the geometry, or aspects of the geometry, of alluvial channels. When formulating these models it was recognised by the authors that the fluvial system is indeterminate and that an additional relation is required to close the problem and to obtain a solution.  An additional relation is often proposed as an “extremal hypothesis” whereby a selected channel parameter is either maximized or minimized. The term extremal hypothesis is synonymous with variational principle. The principal extremal hypotheses which have appeared in the hydraulics literature 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 channel  width 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) the Minimum Energy Dissipation Rate (MEDR) hypothesis (Yang and Song., 1979) which states that the rate of energy dissipation, yQS  +  y Q S (expressed here per unit channel  length) is minimized; and (iii) the Minimum Unit Stream Power (MUSP) hypothesis (Yang, 1976) which states that the stream power per unit weight of water, US, is minimized. 24  In the foregoing y and y are the unit weights of water and sediment,  Q  is the volumetric  sediment transport rate, and U is the mean velocity.  MEDR is the most general of the three minimization hypotheses. MSP is equivalent to MEDR if  Q << Q, which is generally the case for natural rivers, especially gravel rivers. MIJSP is a special case of MEDR 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 factorf  It is recognised that general equivalence exists between the hypotheses, particularly in the case of natural 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 hypotheses may not apply. The best example is MSP in the case of imposed  Q and S such as in a laboratory  study where equilibrium is attained primarily through width adjustment. Under these constraints the 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 alluvial channels (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 extremal hypotheses remains controversial, some suggest that an additional physically based relation such as bank stability should be used (comments following Bettess and White, 1987; Hey, 1988), or that the apparent progress in fiuvial hydraulics through the use of extremal hypotheses is an illusion (Griffiths, 1984). In Davies’ opinion (Davies, 1987) the empirical success of extremal hypotheses is “deeply significant” and may indicate the possibility of fundamental understanding of river behaviour. But until the reasons for the success of extremal hypotheses are known this approach to fiuvial hydraulics will remain unattractive. There have been attempts in the 25  references cited above to explain the extremal hypotheses from physical reasoning, but these attempts have been unconvincing for one reason or another.  The optimization model to be developed in this thesis will now be formulated in a qualitative manner. 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 bedload capacity of the rivers using the fhll range of flows and not simply a single representative or dominant discharge. The objective function developed in Section 2.5.3 is equivalent to an “extremal hypothesis”.  2.5.1 Independent Variables The independent variables represent the known, external controlling variables which are inputs to the system. The variables which are considered independent with respect to the channel reach under consideration and include the physiographic, geologic and hydrologic properties of the catchment. These are the geology, relief; climate, runoff vegetation, and the volume and calibre  of the sediment yield.  Additional independent variables are related to the channel boundary, namely the bank vegetation, and bank sediment parameters such as cohesion and friction angle. The valley slope is also considered to be independent over graded or engineering time scales. Although the valley slope can be considered to be a dependent variable over geologic time scales, any significant change in the valley slope requires the removal or deposition of large volumes of alluvium which can only occur over long periods of time.  The channel will be modelled using the steady-state, mean values of the sediment yield and flow duration data, as well as the mean bank stability parameters for the representative channel reach.  26  2.5.2 Dependent Variables  The 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, and bedforms. In this thesis only a simplified channel geometry will be assumed. Secondary currents and the planform variables will not be considered explicitly, however a comparison of the channel and valley slopes gives a measure of the channel sinuosity and therefore some limited information on the planform geometry.  Since the valley slope is assumed to remain constant over engineering time scales, any changes in the channel slope are limited to changes in the sinuosity of the channel. Through adjustments of its sinuosity, a channel can change its slope much more readily as only relatively small volumes of sediment need to be eroded or deposited along the meander bends. This is in contrast to the large volumes that need to be moved in order to change its slope through aggradation or degradation.  2.5.3 Objective Function In the following section the objective fhnction will be developed. Kirkby (1977) states that in a complex 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 this limiting process. The principal processes operating in river channels is the passage of the flows and the transport of sediment. The most efficient cross-section for passage of the flows is a semi circle where the hydraulic radius is a minimum. For a rectangular section the most efficient cross section is that with a form ratio (W/ Y) of 2. However Gilbert (1914) found experimentally that the optimum form ratio for transporting sediment ranged from 2 to 20. Natural channels rarely have form ratios less than 7 or 8, and typically much greater. Furthermore analytical approaches such as White et aL (1982) have been quite successful at predicting the channel width and depth by assuming a maximum sediment transport condition. This evidence supports the view of Kirkby  27  (1977) that it is the transport of sediment, rather than the accommodation of the flows, that is the limiting 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 defined a sediment transport efficiency e:  e  =  sediment transport work rate available power  (2 2)  The power available for transporting the sediment can be developed from Fig 2.3. Consider two reservoirs A and B separated by a vertical distance AZ. In the upper reservoir A is a volume of water and sediment, V,., and V, respectively. The total potential energy E, expended in moving the water and sediment from A to B is the sum of the potential energy expended by the water E, : 5 and the sediment E  (2.3)  AZ E=yVAZ+y V 5  where y and y  =  (2.4)  the unit weights for water and sediment. For uniform flow along a channel of  length L, the kinetic energy of the fluid and sediment is constant. The power available is equivalent 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:  available  yVAZ power= TL  28  +  AZ y V 3 TL  (2.5)  which for steady flow conditions can be written as:  (2.6)  available power =yQS+yQS  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 by immersed 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 the velocity of the transported sediment. For bedload K = dynamic friction coefficient, for suspended load K  =  ratio of the fall velocity to the mean sediment velocity.  Therefore substituting Equations (2.6) and (2.7) into (2.2):  e=  —r)Q yQS+yQS K(y  For natural rivers, particularly for gravel rivers  (2.8)  Q<<Q, therefore Equation (2.8) can be simplified  to:  e=  y)Q K(r — 8 QS 7  (2.9)  In alluvial rivers the numerical value of e is typically very small as most of the energy in a river is expended overcoming the frictional resistance to the flow.  29  It will now be shown that the maximization of e is equivalent to the principal extremal hypotheses described in the previous section. The denominator of Eqn (2.8) is the energy dissipation rate of Yang et a!. (1981) per unit channel length, Similarly the simplified denominator in Eqn (2.9) is the total stream power (Chang, 1980). For systems where  Q is independent,  maximization of e  implies minimising the energy dissipation rate and total stream power, and is therefore equivalent to the MEDR and MSP hypotheses. Furthermore where both  Q  and  Q  are independent, the  maximization 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, a  maximization of e implies a maximum value of  which is the MTC hypothesis. Therefore the  various 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 general  with the MEDR, MSP, and MTC hypotheses.  For simplicity Equation (2.9) will be reduced to:  Gb =pQS 71  where Gb  =  (2.10)  the dry sediment transport rate in mass units, and p  =  the density of water which is  also in mass units. The values ofK and (y ‘y) I y are generally constant for a given sediment. For -  bedload sediment K  tan 30°  =  0.58, and (y y) I y  approximately equal to one, and therefore  -  i  1.65. The product of K times (y y) / y is -  is an index of the system efficiency which is not only  proportional to e, but for bedload sediment is similar in numerical value.  30  The value of  is always much less than one for rivers. Eqn (2.10) will be modified in Chapter 7  to apply to variable flows.  2.5.4 Constraints The various constraints on the solution will now be discussed qualitatively. The mathematical formulation specifically pertaining to gravel-bed rivers will be presented in Chapters 3 to 5. The constraints in the optimization formulation are discharge, bedload, bank stability, and the valley slope.  The discharge constraint represents the flows that are available to form the channel and transport the sediment. Included in the discharge constraint is the flow resistance. The roughness of the channel determines the velocity and depth of flow which has implications for sediment transport and bank stability.  The bedload constraint represents the imposed bedload that must be transported by the flows. It is assumed that the channel geometry develops in part as a response to the imposed bedload. All sediment less than 2 mm in diameter is excluded from the analysis as the finer sediment is thought to travel essentially is suspension, and therefore does not directly influence the channel-forming processes. For true equilibrium all grainsizes must be transported through the representative reach at just the rate that they are supplied.  Stable channel banks are an additional requirement for an equilibrium channel. This constraint has not been addressed in previous optimization formulations (Chang 1980, Yang et a!. 1981, and White et a!. 1982). The bank stability constraint will be formulated for noncohesive and cohesive channels.  31  The valley slope constrains the optimal solution as the channel must develop a slope which is less than or equal to the valley slope. The maximum channel slope occurs when a straight channel develops along the valley axis.  2.6 SUMMARY  The alluvial channel has been described as an indeterminate system. Experimental results and numerical investigations indicate that an optimal hydraulic geometry can be observed in fluvial systems. It is assumed that a natural river will tend to adjust to this optimal geometry. This  assumption forms the basis for the optimization model. The equilibrium hydraulic geometry is considered to correspond to a solution where the discharge, sediment transport and bank stability constraints have been satisfied, subject to the condition of maximum sediment transport efficiency.  The solution obtained by the modeling represents the steady-state equilibrium geometry. The transient stages of adjustment will not be considered directly in this thesis. The discussion of the processes whereby a natural channel achieves the optimum configuration will be left until Chapter 7.  32  I  /  Change in Storage  =  i.iS  /  0  Figure 2.1. Definition sketch for a representative channel length which is the length of channel along which the mean channel geometry is approximately constant.  33  (a)  Solution  Curve  Gb  Width  (b)  Solution Curve  s  Width  Figure 2.2. A schematic representation of the optimal geometry in fluvial systems obtained from numerical analysis. (a) Constant discharge and slope (eg Gilbert, 1914). (b) Constant discharge and sediment load.  34  A vw  AZ  B  Figure 2.3. Definition sketch for development of the coefficient of sediment transport efficiency  35  CHAPTER 3 DISCHARGE CONSTRAINT  3.1 INTRODUCTION The discharge is a constraint on the channel system as it represents virtually the total energy  input. The channel development and sediment transport must be accomplished with the available flows. In this thesis the discharge constraint will refer to the condition that the channel must 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 resistance equations will be developed for gravel-bed rivers. The morphological and hydraulic significance of 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 cross referencing between Chapters 3 and 4 is necessary to avoid repetition.  3.2 FLOW RESISTANCE The following discussion will be limited to fully rough flow in channels with hydrodynamically rough boundaries, where the mass sediment transport rate is very small in comparison to the mass discharge rate of water. It is generally assumed that flow in open channels obeys the Prandtl  -  von Karman logarithmic velocity distribution. Flow resistance is often expressed by  any of the following equivalent equations:  u  =  (R J 6.25+5.75 7 Iog(, -  i.1*  “S  36  (3.1)  1 (12.2Rh 3 . 2 y= l ogJç O k,  (3.2a)  J  /  f  (12.2RhJJ 1) f=2.O3lo kS  where U = mean velocity; U  =  (3.2b)  (g Rh S)O.5 which is known as the mean shear velocity;! = the  Darcy-Weisbach friction factor; Rh  =  the hydraulic radius; S = the energy slope which is equal  to the longitudinal channel slope under uniform flow conditions; g  =  gravitational acceleration;  and k 3 = the roughness height which is the equivalent sand roughness.  Equation (3.1) was developed for open channel flow by Keulegan (1938) from the work of Nikuradse (1933) on flow resistance in closed conduits. Equation (3.2) is commonly known as the Colebrook  -  White flow resistance equation. The Colebrook White form as presented in -  Eqn (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 coefficient value 12.2 has been obtained experimentally. The roughness height k 3 was originally defined by Nikuradse (1933) and Keulegan (1938) as the diameter of uniform sand grains glued to the pipe or channel perimeter. Keulegan (1938) found that Equation (3.1) fitted the experimental data of Bazin (1865), and that k 5 was approximately equal to the mean grain diameter of the sediments that had been used to line the experimental trapezoidal flume.  Natural gravel rivers are typically characterised by a coarse, graded, bed surface or armour layer. Logically one would assume that ic, would take a value close to the mean grain diameter of the bed surface. However several investigators have concluded that D 50 or even a coarser  37  3 has . The following relation for k 3 characteristic grain diameter underestimated the value of k been proposed:  (3.3)  CD k = 3  where C is a constant corresponding to D. The values of C, were obtained by fitting experimental or flume data to Equations (3.1) or (3.2). There is a wide range of values for C from Ic  =  3 1.25 1)35, (Ackers and White, 1973) to k  =  3.5 D , (Chariton et a!., 1978; Bray, 9  1979, 1982a). Other examples from both field and flume studies are 3 = 3.5 0 (Bray, 1979, 1982a), and k D 1974), k = 6.8 5  =  0 (Kamphuis, D 2 9  (Hey, 1979; Bray, 1982). The value  3 is composed of the grain, form, and planform roughness which are discussed below. of/c  Fig (3.1) shows the friction factorf from the field data of Chariton et al. (1978), Bray (1979), . The 50 Andrews (1984), and Hey and Thorne (1986) plotted against relative roughness, Rh ID mean depth  Y is generally published and used rather than  Rh, however for these natural  channels with width/depth ratios generally greater that 10, this introduces little error. The points represent the values at the bankfill discharge with the exception of Bray (1979) who gives the 2-year flood which generally corresponds to near bankfiill conditions.  50 was obtained by minimizing the sum of the squares of the errors The optimum value of C between the observed and calculated values off The calculated value off were obtained from Eqn (3.2b) the value of k given by Eqn (3.3).  50 was selected as the characteristic grain size because it is most consistent with the original D 0 ranged between 3 by Keulegan (1938). For the data analysed, the values of C, definition of k 0.40 and 56, and the optimum value was determined to be C,o = 5.8. This compares reasonably with the value of C,o = 6.8 obtained by Bray (1979, 1982a).  38  3 Eqn 3.2 with the optimum value of k  =  5.8 C 50 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 investigations mentioned previously. It will now be shown that this scatter should be interpreted as a systematic displacement off above a limiting value which is due to the grain roughness. The additional roughness is interpreted as the effect of bedforms, bars, other macro roughness elements, and planform irregularities along the channel.  3.2.1 Subdivision off into Grain and Form Components One of the principal contributions of Einstein (1950) and Einstein and Barbarossa (1952) to mobile-bed fluvial hydraulics was the assumption that the total channel shear can be expressed as 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 flow exerts on the channel boundary, the value r’ is the component of the total shear stress due to the grain roughness, while “is the component of the total shear stress due to bedforms and other 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  39  (3.5b)  rM=I_pU2  (3.5c)  wheref’ is the friction factor due to the grains, and!” is the friction factor due to the bedforms and 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 a manner analogous to subdivision of z  The shear stress r is subdivided into the grain and form components here by a subdivision of the friction factor. Einstein (1950) and Einstein and Barbarossa (1952) somewhat arbitrarily assumed that the energy slope S is constant and subdivide Rh into the grain and form components, while Meyer-Peter and Muller (1948) assumed a constant value of Rh and subdivided S. The subdivision of f as in Eqn (3.7) yields values for r’ and r” which are equivalent to the subdivision of 5, but differ from the approach of Einstein (1950) and Einstein and Barbarossa (1952).  50 This is D It is proposed that the grain roughness height, ks’, is approximately equal to . 3 by Nikuradse (1933) and Keulegan (1938) who consistent with the original definition of k 40  developed their equations using data collected from straight pipes and channels with no surface irregularities other than uniform surface roughness.  0 ) 5 50 (which is equal to C For the field data presented in Figure 3.1, the values of the ratio k I D ’ 8 ranged from 0.4 to 56. If the premise that k  =  13 is correct, then the minimum theoretical  3I D 50 is 1.0, which corresponds to the case where the total channel value of the ratio k roughness is due to the grain component only. Of the 176 rivers analysed herein, only 4 had  ’ 3 3 /D 50 less than 0.98. This provides good support for the assumption that k values of k  =  ’ 5 Others have suggested larger values for ks’. Einstein and Barbarossa (1952) used k  =  ’ 3 Parker and Peterson (1980) used k  =  ’ 3 , and Prestegaard (1983) used k 90 2D  =  Tho.  84 (these 3 D  studies are discussed below in Section 3.2.2). If any of these values were correct, then the  50 would be much greater than 1. For example if Parker and 3/D minimum observed values of k 50 would be ’ were correct, then the minimum observed value of k / D 3 Petersons’ value for k 90 / D 50 is typically about 3). expected to be around 6 (as the ratio D  The friction factor due to the grain component in natural gravel channels should therefore be given by:  ( 2 (12.2Rh”Y f’=2.03lo D50  (3.8)  Eqn (3.8) is plotted in Fig 3.1 and it is evident that, with the exception of very few points, this equation forms a tight lower bound to the observed values. Therefore the scatter in Fig 3.1 is not random about a mean value, but is more correctly interpreted as a displacement of the data points above the limiting grain roughness due to the presence of bars, bedforms, and other channel irregularities. The displacement of the data points above Eqn (3.8) in Fig 3.1 is equal to  41  3.2.2 Estimation off” for Gravel-Bed Rivers The division of the flow resistance into grain and bedform components has been widely used in studies 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 bedform resistance in gravel-bed rivers. Three methods for estimating the value of f” will now be reviewed, those of Einstein and Barbarossa (1952), Parker and Peterson (1980), and Prestegaard (1983).  3.2.2.1 Einstein and Barbarossa (1952) Einstein (1950) and Einstein and Barbarossa (1952) argued that the bedform roughness must be expected 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  S 3 ‘PD  the intensity of the grain shear which is defined as:  =  35 pg(s—1)D  (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 grain diameter,  is the preferred index of bed sediment mobility:  42  5O =  (3.11)  50 pg(s-1)D  Unpublished data supplied to the author by R.D. Hey and C.R. Thorne on the sediment size 1.25  30 distribution of the bed surface of several gravel-bed rivers in the UK indicates that D  =  35 and 5 is a reasonable approximation. Therefore the relation between ‘P’D D 3  can be  t*D,o  approximated by:  (3.12)  ‘P  The form resistance coefficient of Einstein and Barbarossa (1952), U/ Us”, is related tof” by the following relation:  =9  (3.13)  The relation of Einstein and Barbarossa (1952) for the form roughness can therefore be  expressed 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) against tD  in Fig 3.2. As Eqn (3.14) was derived by Einstein and Barbarossa (1952) from a limited  amount of data from sand-bed rivers, the poor agreement with the data from the gravel-bed rivers is not surprising. However despite the wide scatter there appears to be a fhnctional relation betweenf” and  t*D,O of the  type proposed by Einstein and Barbarossa.  Assuming a wide channel approximation the value 43  t*D,o  can be expressed as (see Eqn (4.15)):  ,* —  f’ f  RS  —  50 (s—l)D  ,* rD 50  ( 315 )  From Eqn (3.15) it is evident that the relation (3.14) has the potential for spurious correlation as the form roughness, f “, appears on both sides of the equation. In Fig 3.3(a) & (b) the data are subdivided such thatf” is plotted againstf’ / shear stress  t*DJQ.  (f’ + f”),  and against the total dimensionless  It is evident that any functional relation such as Eqn (3.14) is due to the  relationship between f” and f’ /  (f’ + f”),  as the relationship  twf” and  r*D$O  appears  (f’ +f”) a I (a +f”)  totally random. Furthermore the variation inf’ is small compared tof”, thereforef’ / can be approximated by a / (a +f”) , where a is a constant. When the flinctionf” is plotted in Fig 3.3a with a  =  =  0.043, which is the mean value off’ from the data set, this  function describes very accurately the variation betweenf” andf’ /  (f’ +f”),  and therefore it  must be concluded that Eqn (3.14), and therefore the bar resistance curve of Einstein and Barbarossa (1952), is largely a function of spurious correlation, and therefore does not give any meaningful 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 of Meyer-Peter and Muller (1948) whereby S is subdivided, rather than Rh. The flow resistance due to the grain component was calculated using an equation equivalent to Eqn (3.8) except 0 was assumed, and the Chezy coefficient, C, rather thanf was used as a measure of D k’=2 9  the channel roughness. The equation developed by Parker and Peterson for the grain component of the Chezy roughness coefficient, Ci’, is:  44  2 C  f’ (  2 (iir”Y  (3.16)  0 751 JJ 5 8 9 D 2 °  where Y = the channel depth. The notation of the original equation as it appeared in Parker and Peterson (1980) has been changed to make it consistent with this thesis.  An empirical bar resistance curve relation derived by Parker and Peterson relates C’ to The Parker and Peterson bar resistance is analogous to Eqn (3.14), and therefore is as much a result of spurious correlation as the Einstein and Barbarossa (1952) relation. The Parker and Peterson relation therefore gives no meaningfhl estimate of the bar resistance.  In addition Parker and Peterson (1980) conclude that for high discharges close to bankfull the contribution 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 higher discharges, 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 to Meyer-Peter and Muller (1948). To estimate the grain component of the channel roughness, the  4 on the basis that the larger grains D k, was assumed equal to 8 ’ grain roughness height, 5 contributed more to the channel roughness than the smaller grains. The grain slope, S’, was calculated using the following equation:  2 ( u  2 (12.2Y*’Y  4 D 8  JJ  (3.17)  Prestegaard originally expressed Eqn (3.17) in the Keulegan form (Eqn 3.1), however for consistency it has been changed here to the Colebrook-White form (Eqn 3.2).  45  The values of S” were obtained by subtracting the values of S’ estimated from Eqn (3.16) from the observed S. Prestegaard found that the bar roughness accounted for about 60% of the total roughness at Qbf in the rivers studied.  The bar roughness was shown to be independent of grainsize. A reasonable correlation was demonstrated between 5” and the dimensionless bar magnitude, which is the ratio of the bar amplitude 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” or  f  “,  is related to channel morphological features (apart from grainsize), represents a promising  method for estimating the bar roughness. However additional field-based research is required to develop predictive equations.  3.2.3 Variation of f” with Discharge The channel roughness whether expressed as Manning’s n orf is known to vary with discharge in most rivers. For channels where bank vegetation is not a significant contributor to flow resistance, 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 flow resistance 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 was  observed, f’ was calculated from Equation (3.8), and  f” f =  -  f’.  The curve indicating the  calculated value off was determined with Eqn (3 .2b) using the value of k that gives the best agreement between the observed and calculated variation off with discharge. This is discussed below. The data for the Meduxnekeag, Northwest Miramichi, and Big Presque Isle Rivers from New Brunswick are given in Phinney (1975), and the values for the Oldman River from Alberta  46  are given in Kellerhals et a!. (1972, Reach 90). The data for the Oldman River were also used by 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 only  slightly with increasing discharge. This decrease is expected as the relative roughness Rh / ]35 increases with increasing  Q.  A much greater decrease  f” with increasing Q is evident,  and  this has a large influence on the totalf This reduction inf” is consistent with the interpretation by Kellerhals et a!. (1972) and Parker and Peterson (1980) that at low discharges the bars and pool-riffle sequences in gravel rivers contributed greatly to the channel roughness, however for higher  Q  values these features become “drowned out” and contribute less to the total  resistance.  Parker and Peterson (1980) went even fi.irther and concluded that the effect of the bars on channel resistance at the higher flows when the bed becomes mobile are negligible, and the form resistance approaches zero. This appears to occur for the Northwest Mirarnichi River (Fig 3.4(b)) for values of  /sec. However Dr. Dale Bray (Personal 3 in excess of about 400 m  Communication, 1994) has indicated that the Northwest Miramichi River is not highly mobile at this 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 the  Big Presque Isle River the value off” is approximately equal to!’ at high discharges. For the Meduxnekeag and Oldman Rivers the value off” at high  Q  values is less than  f’,  but  nonetheless makes a significant contribution to the totaif As mentioned previously, the data in Fig 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 that  f” approaches zero at higher Q,  and yet as is indicated in Fig 3.4(d) this is not so. The reason 47  ’ 3 for this discrepancy is that the relation of Kamphuis (1974) where k ’ 3 Parker and Peterson, rather than k  =  =  2D 90 was used by  50 used herein. It must be concluded that the D  Kamphuis relation includes roughness elements in addition to grain roughness.  The total channel friction is given by Eqn (3 .2b). For modeling purposes, or for computing synthetic stage-discharge curves it is important to know how the values of k andf vary with discharge. Eqn (3.2b) is plotted as the calculated f curve, together with the values of calculated from Eqn (3.8), and the observed variation off with  f’  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 the observed and calculated values off with emphasis on the higher discharge values because most channel adjustments occur at high discharges.  3 The data from the Big Presque Isle river (Fig 3.4c) indicate that a constant value of k  =  0.28 m  produces a good estimate of the observed variation off The data from the Meduxnekeag and Oldman rivers (Fig 3 .4a, d) indicate that at high values of discharge a good agreement between the observed and calculated values of f can be realised with a constant value of k. However data from the Northwest Miramichi River (Fig 3.4(b)) indicate that the value of k is highly  50 from each of the four rivers variable with discharge. Table 3.1 contains the values of D . 8 together with the adopted value of k  For modeling purposes it is most important to accurately estimate the values of f at high discharges where most of the bedload transport and bank erosion, and therefore channel adjustments, occur. Three out of the four rivers shown here display stage-discharge relations  3 for high discharges. Therefore for modeling which indicate relatively constant values of k 3 which remains constant for all purposes it will be assumed that a river has a single value of k discharges and throughout any channel adjustments including changes in the value of . 50 In D Section 3.4 the total roughness height will be divided into bed and bank components, and values of the separate components will be held constant. It is acknowledged that this is a 48  limitation of the modeling procedure, however this can only be resolved through further research into the sources of the form or bar channel roughness, and how this might be related to the channel geometry. 50 from Phinney (1975) and Kellerhals et a!. (1972) together with the Table 3.1. Values of D adopted value of k for the Northwest Miramichi, Meduxnekeag, Big Presque Isle, and Oldman Rivers. River Muduxnekeag Northwest Miramichi Big Presque Isle Oldman  D 5 0 (m) 0.091 0.08 1 0.064 0.040  50 k /D  k 3 (m) 0.20 0.10 0.28 0.20  2.20 1.23 4.38 5.00  3.3 BANKFULL DISCHARGE AND OVERDANK FLOW The morphological significance of the bankfbll discharge and the implications of overbank flow will be discussed in this section. The bankfitll discharge at a river cross section is defined as the flow that just fills the channel to the tops of the banks. A thorough review of the definitions of bankfull discharge is given by Williams (1978). The floodplain is assumed to be the level of the active floodplain which has developed from recent channel activity. In some instances the valley flat may be equivalent to the active flood plain. A definition sketch of the channel and floodplain is shown in Fig 3.5. The definition of Wolman (1955) will be used and this is the stage where the width to flow depth ratio (W / Y) is a minimum. In the simple case of a prismatic channel as is illustrated in Fig 3.5, the bankfhll discharge occurs when the flow reaches 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 total  depth of flow, and H is the vertical height of the main channel banks, and the depth of flow on the floodplain is equal to Y- H. At the bankfhll stage f3 13>0.  49  =  0, and for discharges in excess of Qbf  For values of the discharge up to the bankfhll discharge stress  t  both increase with increasing  greater than for any lower  Q  Q.  At  Qbf  Qbp  Rh and the mean boundary shear  the mean bank and bed shear stresses are  values. This is important from bank stability considerations  because if the sediment which forms the channel banks is not being eroded at  Qbf  it will be  stable at all lower discharges.  A more complex case exits for flows in excess of Qbf The flow is spread over a wide area and there are complex interactions between the overbank flow on the floodplain, and the channel flow. This floodplain-channel interaction was first demonstrated experimentally by Sellin (1964; although see Cruff (1965) for earlier unpublished reports) who identified vortices with vertical axes at the interface between the channel and floodplain flows. These vortices result in momentum exchange between the relatively fast channel flow, and the slower floodplain flow  and produce additional flow resistance effects.  The simplest case for analysis is the channel with an infinitely wide floodplain. The excess flow above bankfi.ill will be spread across an infinitely wide area, and therefore the flow depth on the floodplain is essentially zero, while the depth within the channel does not increase beyond the bankfull depth. For discharges in excess of the Qbp value of f3 remains equal to zero, and the floodplain-channel interaction can be assumed to be negligible. The bank and bed shear stresses which developed at Qbf therefore represent maximum values which are not exceeded at higher discharges. A channel which has stable banks at Qbf will be stable at all other discharges, and the sediment transport rate also reaches the maximum value at  For 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 interactions between the floodplain and main channel flow. These complex flow patterns have been the focus of a considerable research effort. Several investigators have demonstrated the influence of 50  the floodplain flow on the main channel hydraulics by constructing physical models, measuring the hydraulic parameters of the main channel with floodplain flow, and then isolating the floodplain from the main channel by smooth impervious barriers such as glass sheets.  An  example is shown in Fig 3.6 from Sellin (1964). The isovels for the channel with floodplain flow indicate the mean velocity and discharge within the main channel are reduced very significantly when compared to the same channel with the flood plain isolated from the main channel. From his experiments Sellin (1964) determined that the velocity and discharge in the main channel are reduced by approximately 30 %.  It can be observed from the velocity gradients in Fig 3.6 that the bed and bank shear stresses are also reduced significantly due to the floodplain-channel interaction. Barishnikov (1967) performed experiments similar to those of Sellin (1964) to determine the effect of the floodplain-channel interaction on the bedload transport capacity of the channel and determined that the bedload capacity of the main channel could be reduced by 75 - 80 % due to overbank flow interactions. Similarly Meyers and Elsawy (1975) investigated the effect of the floodplainchannel interaction on the boundary shear stress and found that the mean and maximum values of the bed and bank shear stresses could be reduced by 20 - 30% of the bankfiill values for small positive values of3.  In general, it has been determined that the effects of the floodplain-channel interaction are greatest when the difference between the flow velocities on the floodplain and channel is greatest. The interaction effects increase with increasing floodplain roughness (Barishnikov, 1967; Knight and Hamed, 1984), for larger ratios of floodplain width to channel width (Knight and Demetriou, 1983; Holden and James, 1989), and for small non-zero values of (3 (Knight  and Demitriou, 1983). Several investigators have determined that the floodplain-channel interaction effects on the main channel flow become negligible for values of (3 greater than about 0.3 - 0.5 (Posey, 1967; Meyers and Elsawy, 1975; Bhowmik and Demisse, 1982; Pasche  and Rouve, 1985). 51  In conclusion it is evident that for values of 13 between 0 and about 0.5, the bed and bank shear stresses within the main channel are in general somewhat less than the values at  Qbp  and that a  channel 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 no greater, or even less than at Qbf This conclusion does not necessarily apply to very large flood events where 13 is greater than about 0.5 and the floodplain-channel interactions become negligible.  In the present model formulation the infinite floodplain model will be assumed where (3 remains equal to zero for discharges equal to and exceeding Qbf The values of the bed and bank shear stresses 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  Qbf  they will be assumed to be stable with respect to fluvial erosion of the bank sediment for all other discharges. Furthermore the value of the bed shear stress calculated at Qbf will be used to model the sediment transporting capacity of the channel for flows which exceed Qbf  3.3.1 Recurrence Interval of Bankfull Flow Several investigators in the past have concluded that Qbf corresponds to a characteristic return period. Wolman and Leopold (1957) and Leopold et al. (1964) that an average recurrence interval of 1.5 years based on the annual maximum series is a good estimate of Qbf Similarly Dury (1973) suggests a value of 1.58 years. Nixon (1959) based his analysis on the partial duration series and found a mean recurrence interval of about 0.5 years. According to Henderson (1966, p. 465) the result of Wolman and Leopold (1957) is equivalent to a recurrence 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 for the active floodplain and found a mean recurrence interval of 0.9 and 1.5 years based upon the 52  partial duration and annual maximum series, respectively. A plot of the frequency distribution for the analysis of Williams (1978) is shown in Fig 3.7. Disregard the valley flat curve as Qbfhas been defined herein as the height of the active floodplain. While the mean recurrence values in Fig 3.7 agree closely with the 1.5 year average of Wolman and Leopold (1957) and Leopold et aL (1964), there is a wide spread of values between 1.01 to 32 years based on the annual maximum series (0.25 to 32 years for the partial duration series), and the standard error associated with this mean value is 0.277 log units. Williams (1978) concluded that because of the wide range in recurrence intervals an average recurrence interval has little meaning and is a poor estimate of Qbf  The suggestion of a characteristic recurrence interval has resulted in the widespread assumption that Qbf is a primary feature of the watershed, and can be treated as an independent variable whereby the value of Qbf is imposed on the channel. This has been a prime assumption in several 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 et a!., 1978; Andrews, 1984; and Hey and Thorne, 1986). However if one agrees with the conclusion of Williams (1978) that there is in fact no characteristic recurrence interval for the assumption of Qbf as an independent variable becomes somewhat tenuous. Furthermore the actual value of Qbf is defined by the width, depth, slope, and roughness of the channel, all of which are dependent variables. The optimisation model will be used in Chapter 7 to show that the value of Qbf can be viewed as a dependent variable.  3.4 SUBDiVISION OFfINTO BED AND BANK COMPONENTS The preceding discussion requires the assumption that the values of f for the bed and bank sections of the channel are equal. Alternatively the wide channel approximation may be assumed whereby the channel roughness is equal to the bed roughness only, and the contribution of the bank sections to! is negligible. The distribution of the shear force (per unit  53  channel length) which is defined as the product of shear stress and wetted perimeter can be divided into bed and bank components:  (3.18)  ank 3 bnk’  bed.d  where P is the wetted perimeter, and the subscripts bed and bank indicate the bed and bank components of the total (Fig 3.5b). Eqn (3.18) forms the basis for the empirical method developed by Knight (1981) and Knight eta!. (1984) for calculating the mean values of the bed and bank shear stresses, and is used in Chapters 4 and 5 to calculate the bedload transport capacity and to assess the bank stability.  By using Eqn (3.5) to express  t  in terms of J dividing throughout by P and cancelling like  terms, Eqn (3.18) can be expressed in terms off  ff  J  Pbedf  Pbank  ibank  jbed  wherefbed 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 wide channel approximation can be used, and it can therefore be assumed that the contribution of the bank roughness to the total channel roughness can be ignored. However in the modeling work to be undertaken in Chapters 5 and 6 the sediment transporting capacities and bank stability of narrow channels will also be assessed, and the effect of the bank roughness on the total must be considered.  54  The values  Offbed  a.ndfbank can be calculated with Eqn (3 .2b) using the respective values of ksbed  and ksbaflk together with the appropriate values for Rh. It is an important assumption that the bed and bank friction factors only influence a portion of the total cross-sectional area, and in order to calculate fbed and fbank with Eqn. (3.2b), the bed and bank components of the total hydraulic radius, Rhbd and Rhbk, must be determined. The side-wall correction procedure for flume studies developed by Johnson (1942) and Vanom and Brooks (1952; as reported in Vanoni, 1975, p.l52l5z1) will be used to calculate the bed and bank components of Rh. The principal assumption in this method is that the cross-sectional area, A, can be divided into bed and bank components:  A=Ad+A  where Abed =  =  (3.20)  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 to  be no interaction between the adjacent bed and bank areas, and therefore these areas must be divided by a line of zero shear. Additional assumptions are that the mean velocity and energy gradient 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 bank sections must be lines which pass through the junction of the bank and bed and are orthogonal to the isovels, an idea that is attributed to Leighly (1932). Rather than lines of zero shear, the orthogonals are more correctly viewed as lines of zero net momentum flux. Assuming zero shear along the orthogonals, the only force that is resisting the stream-wise gravitational force of each section is the shear force per unit length of channel acting along the respective channel perimeter, hence:  bank’ank  =yAbS  55  (3.21)  which can be rewritten:  rbank  bank  .  —  Pbank  —  a  7’  Eqn (3.22a) can also be expressed in terms of the corresponding bed values:  1?Jibed  (3.22b)  =-=-  7’ S  ‘b.d  In 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 of the cross-section, the isovel contours must be known. However an alternate approach for estimating Rhbed and  is suggested by Eqns (3 .22a, b). In Chapter 4 empirical relations  Rhbank  developed by Flintham and Carling (1988) are used to calculate  ;ed  and  Tbank  in order to  calculate the bedload transport rates and to assess the stability of the banks. Therefore, the values of Rhbed and Rhbk can be calculated using Eqns (3.22a, b) with the values of rbank  rbCd  and  being obtained from Eqns (4.8) and (4.9).  For known values of bed’  bank’ ksbd  and ksbk, the values of  with Eqns (4.8) and (4.9), and RhbCd and  bed 1 ‘  and  ank  can be calculated  from Eqn (3.22a, b). Once these values are  Rhbank  known fbed and fbank can then be calculated from the following equations which are modifications of Eqn (3.2b): /  fb  =  f  2.O3 lo  /  fb  bank  k Sb  J  (3.23a)  I  f  ‘I (12.2RJi bedjJ lo L2.03 k abed  56  (3.23b)  and the total! then calculated from Eqn (3.19).  3.5 DISCHARGE CONSTRAINT The discharge constraint is defined here as the requirement for a channel to have a discharge capacity at bankfull equal to an imposed or trial value of Qbf The discharge constraint is used to determine the size of the main channel.  The discharge constraint is formulated through the definition of volumetric flow rate as follows:  (3.24)  UA=Qbf  where U is the mean velocity, and A is the total cross sectional area, with both corresponding to the value at the bankfull discharge. The value of U is calculated using the Darcy-Weisbach equation: I8gRS U= j 1  (3.25)  where the friction factor! is given by:  (3.19)  and the bank and bed components off given by:  fb  (12.2 Rh ( =2.O3 lo k  57  -2 bflkJJ  (3.23a)  fbed =  (2.03 lo(12.2 Rh  ‘  bed  k Sbed  JJ  (3.23b)  It will be assumed that the values of ksbaflk and ksbCd are independent variables whose values are known and remain constant during stage changes and channel adjustment. This is known to be untrue, however the current knowledge of channel roughness adjustment is insufficient to account for changes in ksbaflk and ksl,ed  The values  Of Rhbk  and Rhbed are determined from Eqns (3 .22a) and (3 .22b) respectively: A bank  Rh bed  ==--  bed 3 ‘  with the values of Thed and  rbank  c 41 (L  )‘S  .  a  (3.22b)  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 transporting capacity. The flow-duration curve will be used as input, and the discharge constraint modified to calculate the  Tbed  and rbank values for each flow.  3.6 SUMMARY  The discharge constraint has been developed in this chapter. This constraint is composed of  two 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 friction factor is subdivided into grain and form roughness. The friction factor due to the grain roughnessf’, is calculated from the Colebrook-White equation with the roughness height equal 50 (for wide channels). No relationship describing the form to the median bed grain size, D 58  roughnessf” has been determined. For many gravel rivers the value off” is significant, and may be greater than f’, even at high in-bank flows. It is demonstrated that the bar resistance curve of Einstein and Barbarossa (1952) is largely a result of spurious correlation and therefore does not return meaningful values off”.  The wide channel approximation is valid for most engineering purposes, however the modeling to be undertaken in this thesis requires that the friction factor be calculated for narrow channels. In order to calculate the total friction factor the channel cross-section is subdivided into bed and bank sections as suggested by Einstein (1942) and Johnson (1942) with the aid of the empirical boundary shear relations of Flintham and Carling (1988). Roughness values are assigned to the bed and bank portions of the channel, and the friction factor of each section is obtained 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 a  dependent variable, rather than as an independent variable as is commonly assumed. The optimisation model will be used to test this idea in Chapter 7.  59  Hey and Thorne Andrews Chariton et al. Bray (1986) (1984) (1979) (1978)  A  *  0  Equation (3.2b) k= 5.8 050  0.6  Equation (3.8) 50 k= D  *  f  0.4  0 A 0 A  * LI 0  E  0.2  0  L)  0  5  10  20  50  100  Rh/DSQ  Figure 3.1. Variation of f with respect to Rh / D 50 for selected gravel-bed rivers for bankfull or near bankfull conditions.  60  0.7 Hey and Thorne Andrews Charfton et at. Bray (1979) (1978) (1984) (1986) A  0.6  *  0  Equation (3.14) Einstein and Barbarossa (1952)  0.5 **  0.4  f.. 0.3  *  A 0  A  0.2  0  *A  ) D  0.1  D  0 0  I  I  I  I  0.01  0.02  0.03  0.04  0.05  0.06  0.07  50  Figure 3.2. Variation off” with respect to bankfull or near banlcfull conditions.  zI*D,O  61  for selected gravel-bed rivers for  0.7  (a)  0.6 0.5 0.4 0.3  0.2 0.1 0 (0.1)  f’I(f’ +f) 0.8  (b) 0.6  f’  0.4  0.2B  B 0 00  0 0  0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16  *  50 D Hey and Thorne Andrews (1986) (1984)  Chariton et al. (1978)  Bray (1979)  0  *  0  Figure 3.3. Variation off” with respect to the components of z*DSO for selected gravelbed rivers for bankfiull or near bankfull conditions. (a)f’ If (b) rD . 30  62  (a)  0.25  0.2  (b)  0.2  0.15  0  0  C 0  C 0  •t5 0.15 0.1  0.1 I  0.05 00  (c)  f 0.05  0  100 200 300 400 500 600 Discharge, Q (me/sec)  200 100 300 400 Discharge, Q (me/sec)  0.2  0.15 0  0  13  C  0  13 0.  C 0  13  13  I.  0.05  50 100 Discharge, Q (m /sec) 3  Discharge, Q (m /sec) 3  f observed  calculated  0  ----r---  f calculated --e-  Figure 3.4. Variation of f with respect to Q. (a) Meduxnekeag River. (b) Northwest Miramichi River. (c) Big Presque Isle River. (d) Oldman River. The values of k used to calculateffor each river are (a) 0.20 m, (b) 0.10 m, (c) 0.28 m, (d) 0.20 m.  63  (a) w  \i  FLOODPLAIN  MAIN CHANNEL  FLOODPLAIN  (b)  w  °°bank  bed  Figure 3.5. Definition sketch of prismatic channel. (a) Composite channel including floodplain. (b) Main channel only.  64  Figure 3.6. Experimental velocity contours (from Sellin, 1964). The upper figure indicates the velocity contours with overbank flow, and the lower with the floodplain isolated from the main channel.  65  RECURRENCE INTERVAL. IN YEARS (PARTIAL DURATION SERIES)  RECURRENCE INTERVAL. IN YEARS (ANNUAL MAXIMUM SERIES)  Figure 3.7. Recurrence interval for Qbf (from Williams, 1978). In this thesis the active floodplain values are of principal concern. The active floodplain data was collected from 36 rivers, and the valley flat data from 26 rivers.  66  O.5A bank  O.5A bank vj  normals isovels  Figure 3.8. Subdivision of cross-section into bed and bank sections. These subareas correspond to the areas of flow that are affected by the bed and bank roughness.  67  CHAPTER 4 BEDLOAD CONSTRAINT  4.1 INTRODUCTION The bedload constraint is a key component of the model and ensures that the amount of sediment that is imposed on the channel can be transported without appreciable net deposition or scour. In Chapter 2 it was argued that the transport of sediment is the limiting channel forming process. Since the model is being formulated for gravel-bed rivers it will be assumed that it is only the transport of coarse gravel bedload sediment that is the limiting process; the finer 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 the bedload transport algorithm.  4.2 BOUNDARY SHEAR STRESS DISTRIBUTION The boundary shear stress will be used to calculate the bedload sediment transport rates, and to assess the stability of the channel banks. Under conditions of uniform, steady flow the mean boundary shear stress z, can be represented by  r=rRhs=pU2  Eqn (4.1) can also be expressed in terms of grain and form components as in Chapter 3.  68  (4.1)  The boundary shear stress is not uniformly distributed across the wetted perimeter, and the local value of the boundary shear stress on the bed and banks can vary significantly from  Several mechanisms are responsible for the non-uniform boundary shear stress distribution and include the lateral transfer of longitudinal momentum, secondary currents, and the non-uniform distribution of roughness elements across the channel perimeter. The influence of these mechanisms and the various approaches to evaluating the boundary shear stress distribution will be discussed below.  4.2.1. Cross-Channel Momentum Transfer  All shear flows are characterised by a net transfer of momentum from regions of high  momentum to regions of low momentum. In rectangular channels momentum is transferred towards the bed and laterally from the center of the channel towards the banks (Cruff 1965). In 2-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 lateral momentum transfer is proportional to the cross-channel velocity gradient (Lundgren and Jonsson, 1964). The velocity which result purely from momentum diffusion in turbulent flows are indicated schematically in Fig 4.1(a).  Superimposed upon the diffusion of momentum is the advective transport of momentum by secondary currents. Einstein and Li (1958) proved theoretically that secondary currents are present 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 of channel curvature. The strong secondary currents can be neglected in straight channels. The effect of the weak secondary currents on the velocity distribution are indicated in Fig 4. lb. Note the depression of the maximum velocity contour below the free surface.  69  4.2.2 Analytical Solutions  Solutions for the distribution of the boundary shear which neglect the influence of secondary currents have been determined for simple cases. Leighly (1932) assumed that the orthogonals to the isovels represent lines of zero shear, and if the isovels are known, the boundary shear stress can be calculated by:  rdP=yS  where  Tdp,  (4.2)  dA and dP are respectively the local boundary shear stress, the area, and the  perimeter bound by two orthogonals (Fig 4.2a).  The area method of Leighly (1932) is of limited practical use as the velocity distribution must be known in order to locate the orthogonals from which dA and dP, and hence dz can be determined.  In a summary of the work of the USBR, Lane (1955b) gives the results of some analytical and numerical studies to determine the values for the maximum bed and bank shear stresses for rectangular and selected trapezoidal channels. The solutions were obtained by assuming a power, rather than logarithmic, velocity distribution along lines normal to the boundary. The results 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 yYS and 0.76 yYS respectively.  Lundgren and Jonsson (1964) used Keulegan’s (1938) assumption that the velocity distribution is logarithmic along lines normal to the boundary. They developed the modified area method which computes the value of  ip  bound by two normals to the boundary (Fig 4.2b). Since the  normals to the boundary do not represent lines of zero shear, the shear stress along these  70  normals must be considered. This shear stress is equivalent to the lateral diflhsion of momentum across the normals.  The modified area method of Lundgren and Jonsson (1964) can be expressed: dA  (4.3)  TdP_Y’p_P6t  where  TdP,  dA and dP are bound by the normals to the boundary, and e is the coefficient of  lateral 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  from  Equation (4.3) agrees well with the method of Leighly (1932). The drawbacks of this approach are that secondary currents are neglected, and that it is applicable only to channels with gradually curving profiles. This method cannot be used for rectangular or trapezoidal cross sections.  4.2.3 Experimental Methods Given the complexities of the effects of secondary currents and the non-uniform distribution of roughness elements across the channel boundary, several investigators have approached the problem of estimating the mean or maximum bed and bank shear stresses by formulating empirical relations based on experimental data.  In a number of investigations velocity distribution or Preston tube measurements in rectangular and trapezoidal flumes have been used to determine the distribution of the boundary shear stresses (Cruff 1965; Rajaratnam and Muralidham, 1969; Kartha and Leutheusser, 1970; Gosh  and 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 et  71  aL (1984), with the subsequent modifications by Flintham and Caning (1988) will be used in this thesis to determine the values of the mean bed and bank shear stresses,  Tbed  and  ;ank  This  method is based upon the distribution of the total shear force SFotai into bed and bank components:  =  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 bank components. The channel is defined in Fig 3.5.  The proportion of the shear force acting on the banks SFbaflk is given by:  —  AL  bank  bank  Velocity 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 was initially developed by Knight (1981) for smooth channels using his own data, and the published data from Cruff (1965), Gosh and Roy (1970), and Meyers (1978). The general form of the relation was verified in Knight et a!. (1984), and the values of the coefficients refined with additional experimental data collected by the authors, together with data from Noutsopoulos and Hadjipanos (1982) and Kartha and Leutheusser (1970).  72  Knights 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 by correlating the ratio of the bed to bank perimeter lengths  bed  /  ‘bank’  rather than the aspect  ratio W/ Y. Flintham and Caning included their own data from a trapezoidal channel, together with the results of Gosh and Roy (1970) to develop the following equation for SFba, presented here in a power form, which is a modification of the empirical equation of Knight (1981) and Knight etaL (1984):  =  1.766  (1d e  +  1.5  (4.7)  ban/c  The values of ;ed and  tbaflk  can be estimated from the following equations which are algebraic  manipulations of Equations (4.5) and (4.6):  I(w+p  bank_SF banlc[  yJ’S  )sin8l  4)7  2 +0.5] =(i-si)[  (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 against the ratio W / Y in Fig 4.4. A comparison with Fig 4.3 shows that the results are reasonably consistent with the results of Lane (1955b).  4.2.3.1 Non uniform bed and bank roughness The effect on the shear stress distribution of the non-uniform distribution of roughness elements across the channel has long been recognised. Einstein (1942, 1950) allows for the influence of 73  bank resistance on the bed shear stress when calculating the sediment transport rates. Following from Einstein’s work, Johnson (1942) developed a side wall correction procedure to be applied to laboratory studies of bedload transport which were being undertaken in narrow glass (or other material) walled flumes. Knight (1981) analysed SFbaflk for channels with different bed and bank roughness values and developed an empirical correction relation for different bed and bank roughness. This correction relation is a function of the ratio of the bed to bank roughness heights, ksbed /  ksbank  Flintham and Carling (1988) performed additional experiments with  differential bed and bank  roughness and developed the following equation which is a  modification of Eqn (4.7):  JFbJ  (P bed i I i’ D  “ 1 bank  8 bed /ksbank• The coefficient C is a function of the ratio of the roughness of the bed to the banks, k 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 from experimental values of ksbed /  ksbank  between 1 and 91.1. This function was forced to pass  through the point C = 1.5 for the value of the ratio  (  C=1.5kdJ  / ksbank = 1:  (4.11)  The influence of the ratio kSbed / kSbank on the mean bed and bank shear stress values is indicated in Fig 4.5. As  ksbed  / ksbank increases, a greater proportion of the total shear force is carried by  the bed and the value of  ed  increases, together with a concomitant decrease in the value of r  74  bankS  The value of  rbank  is more sensitive to the ratio ksbed/ ksbk than  Tbed.,  especially for large  values of WI I’.  4.2.4 Grain and Bar Shear Einstein (1950) and Einstein and Barbarossa (1952) proposed that the total shear could be divided into grain and bar (form) components:  (3.4)  where, as in Chapter 3, the superscripts’ and” refer to the grain and bar or form components respectively.  Eqn (4.5) can therefore be rewritten in terms of the grain and form components:  = (ned + Tbed) ‘,ed  For a wide channel approximation  F, and  bed  +(r  bank 3 ‘  +  (4.12)  <<F, and Equation (4.12) can be  simplified to: V=  Einstein (1950) has argued that only r  ‘bed  (4.13)  + Vbed  contributes to the bedload transport. By definition  (see Chapter 3): V’=pU2  (3.5b)  Now: = 2 U  8gRS  f  Combining Equations (3.5b) and (4.14) yields: 75  8r =  f  (414)  (4.15)  f wheref 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 Figs 3.1 and 3.2.  Fig 4.6 indicates that the value of  r’  is bounded by z, and the scatter below the upper bound  indicates that for most channels the value of ‘is significantly less than zThe exception is a few channels where r’ appears to exceed r which it cannot by definition. This anomaly is attributed to measurement errors as in practice it may be difficult to assign truly representative values of 50 Rh etc to a river reach. D ,  For these data the mean value of the ratio  t  ‘i r is 0.49, and the minimum value is 0.11. This  indicates that for some channels as little as 11% of the total bed shear stress is available for bedload transport even at or near  Qbank.  This contradicts Parker and Peterson (1980) who  conclude that for high in-bank flows, the total boundary shear stress is available for bedload transport. This result indicates that large variations in the calculated bedload transport rates can result if Tbed rather than r  bed  is used.  Despite this result, the Parker (1990) surface-based bedload transport relation, which uses to calculate the bedload transport rates and not r  ‘bed.,  ia  will be used in a modified form in  Chapter 7 of this thesis to calculate the sediment transporting capacity of the modelled river channels. In order to reformulate the Parker (1990) relation on the basis of r ‘bed, considerable effort is required.  76  4.3 MODELLING SEDIMENT TRANSPORTING CAPACiTY AT BANKFULL Optimization 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 rate corresponding to the dominant or bankfull discharge. Similarly Hey and Thorne (1986) included the bankfI.ill sediment transport rate as an independent variable in their regression analysis. This has been justified as most of the transport, particularly for gravel-bed rivers, typically occurs at or near bankfull. However it will be argued that the duration of the transporting flows must be also considered, and that the sediment transporting capacity should be defined by the total bedload which can be transported over a significant duration T (eg. 1 year), and not by a single transporting rate.  Consider two channels which have identical values for both  and the sediment transport  capacity at bankfull Gbf Furthermore assume that these channels are only just above the critical threshold for bedload transport at  so that essentially no bedload transport occurs for  discharges less than Qbf Andrews (1984) determined from 24 gravel-bed rivers in Colorado that duration for which the Qbf was equalled or exceeded ranged between 0.12% and 6%. Therefore the total bedload transported over duration T by two channels with identical Qbf and Gbf values could 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 the channel banks should be assessed at this value, it is not sufficient to model the sediment transport capacity of the channel at the single value of Qbp but rather the flow durations must also be considered. The complete flow-duration data will be used as input to the model. These data are a representation of the complete range of flows over a period of record. For computational purposes the flow-duration curve will be discretized into m segments (Fig 4.7) of quasi-steady, uniform flow. The valuep 1 is the probability of flow Q , where: 1  77  (4.16)  =1.O 1 p  4.3.1 Generalized Equations for Mean Bed and Bank Shear Stresses  The following equations are necessary to calculate the value of Tbedj which is in turn required to calculate the bedload transport rate for each flow  Q,.  The equations used in the discharge  constraint (see Sect 3.6) are modified to calculate the flow geometry and the mean bed shear stress values which correspond to each flow Q.  The continuity requirement for uniform flow is given by:  =Q UA 1  where U 1 is the cross sectional area for flow 1 is the mean velocity, and A  (4.17)  Q•.  The value of U 1 is  calculated using the Darcy-Weisbach equation:  I8gR. S  f.  (4.18)  which is a modification of Eqn (3.25). The values of Rh, andf now take different values for each value 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 given by:  78  fb.  ( (12.2RhbJJ =2.03 lo  =  The values of  (4.20a)  )  Sçj.J  IT  (122 R hbed 1 [2.03 lo k bed I  and Rhbed. are determined from Eqns (4.21a) and (4.21b) respectively:  Rhbed.  Abed =  —  bedj 1  —  —  Rhb,?,(.  The values of 1 •bed• and  Tbankj  =  —  (4.21 a)  S  V  Tbank.  (4.2 ib)  yS  are given by:  Tbank  ygS  =SFbanki[  Tbed.  yYS  The value of SFbaflkj for each flow  j  4y  =(1—sFbank,  JEw  +0.5  ,.  (4.22)  (4.23)  bed  Q, is given by: y1.4o26  S],  ‘ied  I\C]Lk.  where the value C is given by Eqn (4.11).  79  +  (4.24)  4.4 BED SURFACE  Gravel-bed rivers typically develop a layer on the bed surface that is significantly coarser than the underlying material. This layer is one grain thick and is often termed the pavement or armour layer (Kellerhals and Bray, 1971; Parker et aL 1982), and the underlying finer sediment is called the subsurface, subarmour, or subpavement material. In this thesis the term armour will generally be used to describe the surface layer, and subarmour to represent the subsurface material.  The armour layer plays an integral role in the channel dynamics of gravel-bed rivers. The coarse grains which comprise the armour produce the grain roughness of the channel bed and therefore contribute to the flow resistance. Furthermore the armour layer shields the finer subarmour sediment 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 important factor in limiting the availability of sediment and controlling the relationship between streamfiow and bedload discharge.  The grain size distribution of the armour layer can be most simply represented by a single 50 or the geometric mean grain D characteristic grain size such as the median grain diameter ,  diameter D, together with a measure of the grain size dispersion about the mean or median value such as the standard deviation based on the sedimentological phi scale a, or geometric standard deviation a.  The degree of armour development will be defined by the following relation: D 50  80  (4.25)  Where d 50 is the median grain diameter of the subarmour sediment. The value of  350  has been  observed to range from 1 with no armour development, to greater than 6 which indicates a large degree 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 been observed both experimentally and from field observations. For instance Kellerhals (1967) demonstrated the coarsening of the armour layer under steady flow conditions when the supply of bedload sediment was cut off This coarsening of the armour is commonly observed downstream 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 bedload transport rates tend to have values of Sso which approach 1. Dietrich et a!. also noted that the effect of reducing the sediment feed to a channel was for the channel to increase 6. It is 50 and Dsg, as well as o or 0 therefore evident that both D g’ must be considered to be 50 or d, when no 50 and Dsg can take minimum values of d dependent variables. The values of D , which is the maximum 1 surface coarsening is developed, and theoretical maximum values of d subarmour grain diameter. This can be stated formally as:  50 d  100 d  (4.26)  1 dsg Dsg d  (4.27)  50 D  Eqns. (4.26) and (4.27) are termed the armour grain size constraints. In reality the upper bound 90 rather than , 100 as sufficient sediment of this grain d 50 and D is probably closer to d to D diameter must be present in the subarmour in order to form an armour layer.  50 and D are treated as dependent variables which can adjust In the proposed formulation D within the bounds given in Eqns. (4.26) and (4.27). The values of o and og will also be treated 81  as dependent variables. The grain size distribution of the subarmour sediment will be assumed to be equivalent to the long-term grain size distribution of transported bedload sediment (Parker et 100 as well as other subarmour grain sizes are a result of the d a!. 1982). The values of , 50 d, , d  catchment geology and sediment sorting and abrasion which occurs upstream, and therefore these values are imposed on the channel reach, and are therefore considered to be independent variables.  To account for the effect of the armour layer in the model the surface-based bedload relation of Parker (1990) will be used. In this bedload relation Parker explicitly accounts for the influence of the armour layer on the bedload transport rate. The armour layer grain size, which will be 50 is permitted to adjust together with the D represented by ,  b’ ban1,  S, 8, and other  dependent variables until an optimum solution is attained. The Parker surface-based bedload transport relation is presented in a Section 4.5.  4.4.1 Influence of the Armour Layer  The simplest sediment transport model which accounts for the influence of the armour employs the concept of an armour threshold. At discharges (or shear stresses) below the threshold, the armour layer is essentially immobile, and the bedload transport rate is close to zero. Any material which is transported above the immobile armour layer is considered as suspended or sandy throughput load which is considered to be more a function of upstream supply, rather than the local channel hydraulics. Only the load which is considered to result from an exchange between the bed and the flow will be considered in this paper. For many gravel-bed rivers this applies to sediment coarser than about 2 to 5 mm (Parker, 1990).  For flows exceeding the threshold, the armour becomes mobile, exposing the finer subarmour to the flow, and sediment transport rates increase dramatically as shown in Fig 4.8(a). The data from Fig 4.8(a) was collected from Oak Creek, Oregon, USA by Milhous (1972). The critical discharge is approximately 1 m /sec which for Oak Creek corresponds to a value of 3 82  that  is approximately equal to 0.03. Note that a threshold is evident only in Fig 4.8(a) which is plotted with a linear Y- axis. When the same data are plotted in Fig 4.8(b) with a logarithmic Y axis no threshold is evident.  Two models of armour behaviour at discharges above critical are recognised. The conventional model assumes that when the discharge is significantly greater than the critical value the armour breaks down and the subarmour sediment is fuily exposed to the flow. As the discharge wanes on the falling limb of the flood hydrograph the armour layer reforms as the coarser particles become increasingly immobile, while the more mobile fine sediment are winnowed out from the surface layer.  A more recent model has been developed by Gary Parker and others which contends that the armour 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 armour layer was seen to develop which regulated the supply of fine material and resulted in the equal mobility of all size fractions (Parker et a!., 1 982a, b). There is also field evidence that the armour 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 storm events where the bed became mobile, which suggests that the armour layer had remained intact during these events. Andrews and Erman (1986) found that the armour layer of Sagehen Creek in California persisted during an extreme snowmelt event with a peak discharge which exceeded the bankfull discharge by 230%.  The model of Parker for the mobilisation of the armour layer will be assumed in this thesis. The armour layer is regarded as a fundamental feature of the channel geometry which remains relatively constant once equilibrium has been achieved.  83  4.5 PARKER SURFACE-BASED BEDLOAD TRANSPORT RELATION The Parker (1990) surface-based bedload transport relation was selected to model the sediment transporting capacity of the channel as it is the first relation to explicitly account for the influence of the armour layer on the bedload transport. An additional feature is that the relation can be inverted and used to calculate the grain size distribution of the armour layer which must develop 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 surface based relation is the result of a series of papers authored by Gary Parker and others which dealt with the development of the armour layer in gravel-bed rivers and related bedload transport issues (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. Under the low rates of bedload transport typical of natural gravel-bed rivers the coarse surficial armour layer 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 or scour, and hence must have developed a bedload transport capacity which is equal to the rate at which the bedload sediment is supplied to the channel. This applies to the total load, and also to each size fraction. Therefore the coarsest 10% of the bedload sediment must be transported through the equilibrium channel reach at exactly the same rate as the finest 10% of the sediment over 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 2 mm in diameter are considered to be transported as suspended or throughput load and the rates of transport are governed more by supply from upstream, rather than local channel hydraulics (Parker, 1990).  84  Now a prime assumption in the Parker model is that the rate of transport of each size fraction is regulated by the volume fraction of the surface layer that it occupies. The higher inherent mobility of the fine sediment can be countered by reducing the volume fraction of the surface layer which it occupies. Conversely the coarse size fractions become over represented in the surface layer to counter their inherent lower mobility. The net result is that a grain size distribution 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 than the subarmour or bedload sediment grain size distribution. The theory predicts that the armour layer disappears under conditions of high bedload transport rates which is in general agreement with the experimental and field evidence of Dietrich et a!. (1989), Lisle and Madej (1989), and Kuhnle (1989).  The mathematical formulation of the Parker (1990) relation is presented below. The armour and subarmour grain size distributions are subdivided into n intervals, and a representative grain size is assigned to each interval. The proportions of the total by volume which fall within each interval is denoted byf for the subarmour mixture, and 1, for the armour layer.  The basic parameter of the relation is W width of size fraction  which is the dimensionless transport rate per unit bed  normalized per unit volumetric armour layer content (s—1)gq ‘  The value  =  (i)  1.5  (4.28)  ‘  is the volumetric bedload transport rate per unit channel width of size fraction D.  The total volumetric bedload transport rate per unit bed width,  85  qb,  is given by:  (4.29)  b  In order to understand the Parker relation, consider initially a channel composed of uniform sediment of diameter D; for which the value W’ is given by:  (4.30)  0.00218G[  =  where t is the dimensionless bed shear stress obtained by dividing the dimensionless (Shields) bed shear stress by a reference dimensionless bed shear stress  rD  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 following system of equations:  r 54741— 0.853Y 5  c1>1.59  )  G[c1]  exp[14.2(c1_ 1)  —  9.28(— 1)2]  1  1.59  (4.32)  =  42 cF’  IEquations (4.30) to (4.32) will now be generalised to apply to a bed comprised of graded sediment. A key assumption is that the mobility of the armour layer is determined by the mobility of the geometric mean grain diameter D. The function G is calculated as a function of  D. Further modifications are necessary to account for the hiding effects in a graded armour  86  layer, and the relation is generalised to a bed of arbitrary composition by an order-one straining function. The value of W is given by:  =  0 g 0.00218G[cI (o) c] 0  (4.33)  where the function G is unchanged from (4.32), 1 is calculated from Eqn (4.31) with D Dsg divided by a dimensionless reference shear stress  =  0.0386. The function g (4) is a 0  reduced hiding function, and a is a straining function. These two modifying functions will be discussed below.  The reduced hiding function accounts for the hiding of smaller grains and the over exposure of larger grains relative to Dsg. The function  4. is a modification of Eqn. (4.25) and is defined as: D. =_—‘  (4.34)  Dsg = expi1. lnDjJ  (4.35)  Sf  The value of Dsg is given by:  The function g (4) was developed from Oak Creek data and was assumed to have the value 0 (1)1: 0 g go(S.) =  (4.36)  For grains smaller than Dsg the value of g (4) is less than one and the effective shear stress 0 acting on the grain D. is reduced. The converse is true for grains larger than Dsg.  87  The order-one straining function a was introduced based upon an analysis of Oak Creek data to account for the grain size distribution of the armour layer sediment. From a numerical analysis using Oak Creek data the function a was found to vary from an asymptotic value close 0 (and low values of g 3 1 to 1 for low values of c g) for which a coarse, well sorted equilibrium  surface prevails, to 0.453 at high values of  where a finer and more poorly sorted surface  layer is predicted. From this result it was postulated that a generalised straining function co might be a function of  and some measure of the armour grain size dispersion. The measure  of the grain size dispersion utilised is  (4.37)  In the case of uniform sediment  =  0, and co should equal 1. For Oak Creek co  =  . The 0 Co  simplest hypothesis assumes a linear variation in co which is consistent with uniform sediment and 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 Distribution The usual application of the Parker surface-based relation is to input the values of D g, D 3 , arid 1 and then together with  TbedtO  calculate the bedload transport rate per unit channel bed width  which is denoted g, when expressed in dry mass units, and Dsg and o, together with the values of  qb  in volumetric units. However if  are unknown initially, it will be necessary to calculate  the value of these variables for imposed values of Gb. It is important to note here that while the total load Gb is an imposed value, the transport rates gb and q, are in effect dependent variables 88  as their values will be different for different channel geometries. By definition gb  =  Gb / Pbed, and  since the value of Pbed is considered in this thesis to be a dependent variable, the value of gb is also a dependent variable. Furthermore changes in the channel geometry will also affect the values of md which in turn will affect the values of gb and  qb.  By equating Eqns (4.28) and (4.33) and rearranging the following relation for  1,.  is obtained:  (s—i) gqb  =  0.00218 G[sgo  (s a] (Tb, 0 g ) 1  I p)  (4.39)  15  The fraction volume content of the transported bedload sediment in the jth range is denoted which is defined as:  qb•  (4.40)  Combining Eqns (4.40) and (4.39) and rearranging yields:  F  (s—1)gq  —  —  G[sgo  ) 1 go(s  ]  441  5 0.00218(rb / p)’  Since q andf are known for an imposed Gb and a known value of Pbed, the values of F; and the composition 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 channel geometries for which the value of q, is not known in advance. The calculation of an equilibrium armour layer for this more general case will now be considered.  89  For a specific discharge the second term in Eqn (4.41) is constant, although the value of qb may not be known initially. Eqn (4.41) can therefore be simplified to:  (4.42) G[CI)so 1 (3 a] 0 g )  Now by definition the following must hold:  (4.43)  F;.=L0  Therefore Eqn (4.42) can be presented as an equality by normalising over the summation of all  fj/G[(I)sgo  o) a)] (g 0  fj/G[sgo  o) (g 0  wJ  (4.44)  Eqn (4.44) permits the calculation of the values of F. without knowing the value of q,, beforehand.  Since the values of F, and therefore D and  are unknown initially, an iterative procedure is  required in order to solve Eqns (4.31), (4.36), (4.37), (4.38), and (4.44) in order to determine the composition of the equilibrium armour layer, as well as q,. This will be discussed in Section  7.2.5.  By inspection between Eqns (4.44) and (4.41) it is evident that the value of q is given by:  90  O.00218(r  15 ip)  (4.45)  qb =  (s—i) gfj/G[go g (o,) a] 0  4.5.2  Modification of the Parker Surface-Based Relation for Natural Rivers with  Variable Discharge The 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 were performed using single, steady discharges (Parker et a!., 1982b). In its original form it implies that equal mobility occurs for all discharges above the critical for armour mobility, and therefore assumes that the grain size distribution of the transported bedload is equal to the subarmour for all flows in excess of critical. This is known to be incorrect as is indicated by data from Oak Creek (Milhous, 1972) which is shown in Fig 4.11. Note that below the critical  Q  of about 1  50 is approximately 2 mm and represents sandy throughput load moving /sec the value of d 3 m 50 increases with /s the value of d 3 above an immobile armour. For flows in excess of 1 m increasing  Q.  50 of the transported load In its original form the equal mobility hypothesis supposes that d would be a constant for discharges in excess of the critical, and for Oak Creek this constant value would be equal to 20 mm. As Fig 4.11 shows this is clearly incorrect. This was acknowledged by Parker et aL (1982a) who state that equal mobility is at best a crude, but useful, first-order approximation.  However this hypothesis can be correctly applied to natural channels with varying discharge by recognising that for a channel in equilibrium, the constraint of equal mobility must be fulfilled over a significant duration. Parker (1990) acknowledges that the size distribution of the annual  91  yield of gravel is similar to the subarmour. Similarly Church et al. (1991), although referring to fine 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 of an equilibrium armour layer even under controlled conditions requires many hours. The flows in many natural rivers, in particular ‘flashy’ gravel-bed rivers, are probably too variable for an armour layer to develop an equilibrium state with a single discharge. However over a significant duration, which is usually at least one complete water year, a channel which is in equilibrium must 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 supplied to the channel. That is all sediment sizes are rendered equally mobile. Therefore the armour layer which develops in an equilibrium channel represents the grain size distribution necessary to develop equal mobility for the complete range of flows experienced by the channel over a significant duration. However any single discharge is likely to be in disequilibrium with the armour because the sediment size distribution of the sediment being transported by a given flow at any given time is probably different than the subarmour.  is calculated from the equations presented in Section 4.3.2,  For each value  Q,  from which  can be determined. The value  a value of  •bed• 1  is the volumetric sediment transport rate per  unit channel width for flow Q,. Eqns (4.28) and (4.33) must be modified as follows: (s—1)gq,.  (4.46)  = ‘  (r,/p)  ‘  0 1 J =O.OO218G[cF (8 a] 0 g )  92  (4.47)  where  13 qb  values  Vbedj,  is the unit volumetric bedload transport rate for size fractionj for flow , and 1 Io  Co.,  and G  are  not constant, but now vary with  Q,,  and the  Q.  By equating Eqns (4.46) and (4.47) and rearranging the following relation for qb 11 is obtained: O.00218F. qb  =  5 (s—l)gp’  (s) 0 oj g G[ g 3  a]i,  (4.48)  By definition:  =  Pi  (4.49)  s) a] z, g ( v G[goj 0 1  (4.50)  where p is the probability of Q,.  Therefore from Eqn (4.48) and (4.49): O.00218F. (s— 1)g p 15  Rearranging (4.50) together with (4.40) gives the following relationship for 13 which is analogous to (4.41):  F  fJ  —  —  1 p  . g(o) 0 G[  ,]  (s—1)gp q 5 0.002 18  (451)  Since the second term of Eqn (4.51) is constant (although the value is not necessarily known in advance), with Eqn (4.43) the following equation analogous to (4.44) is obtained:  93  f/P F.  =  G[sgoj  o) c]r. g ( 0  (4.52)  ‘‘  o) w} r g ( G[so. 0  By inspection Eqns (4.51) and (4.52), it is evident that the total volumetric unit bedload transport rate per unit bed width q, is given by: 0.00218 qb =  n  (4.53)  m  o) g ( G[sgo. 0  (s- 1)g p’ 5  €]  /sec. 2 If the SI system of units is used, qb will have units of m  4.6 TOTAL BEDLOAD CONSTRAINT The total bedload constraint for the channel subject to the complete range of flows defined in the 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 an independent variable and represents the total load imposed onto the channel from upstream. The value of gb is given by: g=Ispq  where  r is  (4.55)  a constant which is related to the time base of Gb. The value of qb in the total  bedload constraint is calculated using Eqn (4.53). The units of g and Gb must reflect the load imposed over a significant duration. For instance under steady flow conditions such as in a laboratory experiment Gb can have units kg / s as the rate is constant and will be the same for all time scales. In natural rivers with highly variable and seasonal flows the logical time base would 94  be one water year, and therefore Gb would have units of kg / y. In general the modeling to be undertaken 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 takes a value equal to 365.25 24 3600 which is the number of seconds in one year.  4.7 SUMMARY  The bedload constraint has been developed and consists of two components: boundary shear stress and sediment transport. In this thesis, the mean bed and bank shear stress (bank shear stress is necessary to assess the stability of the banks, see Chapter 5) are estimated using the empirical relations of Flintham and Caning (1988). These formulae were derived from experimental data in straight rectangular and trapezoidal flumes, and they permit the estimation of the mean values of  rb€d  and  bank  for trapezoidal channels with different bed and bank  roughness. Flintham and Carling caution that these results may not be confidently applied directly to large scale natural channels or for bank angles less than 45°. This warning is acknowledged, however in the absence of anything better, the relations of Flintham and Caning will 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 are implicitly accounted for in the empirical shear stress relations. The strong secondary currents which are associated with curved channels are not accounted for in this thesis. The secondary currents are considered to exert only a second-order influence on the channel geometry and will result principally in variations of the local geometry from the reach-averaged values. There is some support for this assumption in the regime equations of Hey and Thorne who found that the channel width at pool and riffles sequences varied by less than 5% from the reach-averaged value.  The Parker (1990) surface-based bedload transport relation will be used in this thesis to model the sediment transporting capacity of the channel in Chapter 7. This bedload transport relation 95  was selected because it explicitly accounts for the influence of the armour layer on the sediment transport, and the relation can be inverted to calculate the equilibrium armour layer grain size distribution that is required to transport an imposed bedload for given flow conditions. This 50 to be allows the grain size distribution of the armour layer, together with the value of D treated as dependent variables in the optimization model. Therefore 135 can adjust together with W,  Y, and S to define an optimum channel geometry.  The bedload transporting capacity of the modelled channel is calculated using the complete range of flows represented by the flow-duration curve, rather than at a single discharge such as Qbf  The concept of “equal mobility” is restated whereby the armour layer adjusts such that all  size 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) I  I  I  I  I  I I I  I I I  —  g g  I  f /  ‘  /  —  ‘  I —— —  “ -‘  (b)  V  /  \\\  ‘s:==E —  —  —  —  Velocity Contours Weak Secondary Currents  Figure 4.1. Simplified velocity distribution in a straight open channel. (a) Momentum diffusion only. (b) Momentum diffusion and secondary currents.  97  (a)  V  (b)  dP  Figure 4.2. Area methods. (a) Normals to isovels (Leighly, 1932). (b) Normals to bed (Lundgren and Jonsson, 1964).  98  OwyS’0.75OwyS Q.97OwyS  1.0 0,  0,  Tropezois, z 2  .  V  z I 5  0.7  ‘.‘  C  8  C  /ecton9esI  a.;  1  C 0• —  0.”  ,J.___ ‘if  0  Iv  LAH  ::iz:  0  ‘— .  fl ...l4—  0.7  —  “‘Tropezoids Z  V  ‘Troezods, z2ondI.5—  0.9  0.8 Q  —  02  D  —  [  -  -  T  flTT I  2  3  4  67  8  9  0  _0  b/y  2  .5  7  I 8  —  9 10  b/y  Figure 4.3. Boundaiy shear stress distribution from Lane (1955b). For Fig 4.3 only, b is equivalent to Pbed, y is equivalent to Y, and z is a measure of the bank angle which is equal to cot  99  (a)  1  a)  C  .0 U, C  0.6  0.4  a)  E  c 0.2 C  0  z  0  1  (b)  0.8  0.6 Cu  cr.i C C (0)  C 0)  E  0.2 C  0  z  0  wIY  Figure 4.4. Boundary shear stress variation from Eqns (4.8) and (4.9) for variable bank angle 6 and uniform boundary roughness in a straight trapezoidal channel. (a) Non-dimensional bed shear stress = Vbed / yJ’S. (b) Non-dimensional bank shear stress = Tbank / rYS.  100  (a)  1 0.8  a)  0.6  a)  o 0.4 Cl)  C  a)  E 130.2  e0  z  (b)  0  1  h  0.8  Cl) C a) a) C 0 Cl)  C  a)  U C 0  z  0  5  15  10  20  25  w/Y  Figure 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.  101  140  120  z Cl) C,)  a)  100  80  60  40  20  0  0  20  40  80  60  100  120  140  ) 2 Total Shear Stress, r (N/rn  Figure 4.6. Grain shear stress versus total shear stress. The value of r is computed from the observed channel geometry, and r’ is calculated from Eqn (4.15).  102  C)  a)  E  0 Probability of Exceedence  Figure 4.7. Discretized Flow-Duration Curve. The probability of exceedence refers to a specified time base and period of record. The value Q is the characteristic discharge value assigned to the interval, and Pi is the probability of a discharge being within a particular interval.  103  0.12  (a)  0.1 ‘  -  D  0.080.06  -  0  0.04-  0  D DO  0.02-  D C  0  B I  0  1  2  3  4  3  4  Q (me/s)  (b)  -  0.01 0.001 0.0001 1E-05 1E-06 1E-07 0 1E-08  -  -  -  -  -f -°  2  /s) 3 Q (m  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)  500 400 300  G 200 100 0  0.5  1  1.5  2  1.5  2  •sgO  (b) 100 10 1 0.1 0.01 0.001 0.0001 1E-05 0.5  1 øsgO  Figure 4.9. Variation of the function G from Eqn (4.32). (a) Linear Y-axis. (b) Logarithmic Y axis.  105  2 COo  1.5 —  1  3  2  4  ’sgO 1  with q 5 (from Parker, 1990). See text for  Figure 4.10. Variation of the functions a and discussion.  106  30 0 0  25  0 0  20  d subarmour 00  E  0  0 0 0  0 0  5  0  0002  00E0,0:  3  4  /s) 3 Q (m  Figure 4.11. Variation of d50 1 d with Q for Oak Creek, Oregon, USA. Data from Milhous 50 from the subarmour sediment is approximately 20 mm. (1972). The value of d  107  CHAPTER 5  BANK STABILITY CONSTRAINT  5.1 INTRODUCTION In 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 stability constraint will be formulated for both cohesive and noncohesive bank sediments.  There are two fundamental processes of bank erosion which must be considered: mass failure and the fiuvial erosion of discrete grains or aggregates (Wolman, 1959; Thorne, 1982; Gressinger, 1982). 5.2 COHESIVE BANK SEDIMENT Cohesive channel banks are those composed of clay, silt and fine sand and have typically resulted from over-bank deposition of fine sediment from suspension. Cohesive sediments contain significant amounts of clay minerals which strongly influence the physical properties of the soil. According to Raudkivi (1990, p. 300) when the clay content of a soil is 10% or greater, the physical properties of the soil are dominated by the clay fraction.  5.2.1 Mass Failure Mass failure of a river bank occurs when the driving force FD, which is due to the weight of the failed soil mass w; exceeds the resisting force FR, which is a result of the cohesion c and internal friction angle çS of the soil acting along the failure surface. The unit shear strength of a soil is usually given by the Mohr-Coulomb failure criterion:  108  (5.1)  SR=c+NRtanØ  where SR is the shearing resistance for a unit area of soil, and  is the normal upward force  acting on the base of the soil unit.  Potential failure will occur along the surface where the factor of safety, F,, which is defined by the ratio FR /FD, is a minimum. Failure will occur when F, is less than one. The critical height of a bank  corresponds to F,  =  1, and can be expressed by the following equation from  Terzaghi 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 is the stability factor. Note that N as presented in Terzaghi and Peck (1948) and in Eqn (5.2) is the inverse of Taylor’s (1948) stability number. The value of N, is a fi.mction of  and the  bank angle 6.  The simplest model of slope failure is the Culmann analysis (Taylor, 1948; Spangler and Handy, 1982) which assumes that the failure surface is planar and passes through the toe of the bank (Fig 5.1). This method yields the following analytical solution for N:  N  =  4 sinO cosçS 1—cos(6—.)  (5.3)  The slope of the failed surface, a, is obtained by differentiation:  a= (See Fig 5.1). 109  6+ç 2  (5.4)  Application of the Culmann solution is limited. It can only be applied with reasonable confidence to completely drained slopes close to vertical. For lower angle slopes the failure surfaces are strongly curved and the Culmann analysis under predicts N , and therefore 5 This 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; Morgenstern and Price, 1965; Spencer, 1967) all require trial failure surfaces to be assessed. The friction circle 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 number of vertical slices of convenient width (Fig 5.2(a)), and the driving and resisting forces are resolved for each soil slice (Fig 5.2(b)). The factor of safety is established for that particular surface by summation of the forces acting on all of the slices. A number of trial failure surfaces  are examined until the surface with the lowest value of F 5 is determined. If the value of F 3 is less 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 other methods of slices, the number of unknowns exceeds the number of equations available for solution and therefore some simplifying assumptions are necessary. In the ordinary method of slices the vertical shearing forces between the slices are neglected, and the resultant force on a slice is assumed to act parallel to the base of the slice. This can lead to an under estimation of the factor of safety by as much as 60% (see Nash, 1987). In Bishop’s approach the vertical shearing forces between adjacent slices are considered in the analysis and are assumed to be equal and opposite. A key feature of the Bishop simplified method is an increase in the normal force at the base of each slice which tends to compensate for the side forces on the slices. 110  The solution for the factor of safety using the ordinary method of slices and assuming a circular failure surface in terms of effective stresses is given by:  FR SF  (c’l+(wcosa—crl)tanç5’) wsina  where o- is the mean pore pressure for the slice, and 1 is the basal length of each slice. The soil strength parameters c’ and çS’ are given in terms of the effective stresses. The solution for the . 3 Bishop simplified method is more complex and requires an iterative solution for F  The ordinary and Bishop methods can also be applied to non-circular failure surfaces. Further discussion 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 which require numerical techniques such as finite element analysis for solution (eg. Chen 1975; Chen 5 with & and Lui, 1990), can be used to construct stability curves which show the variation ofN Examples of stability curves are shown in Fig 5.3(a). Those solutions indicated by the solid lines were derived from the graphs and tabulated data published in Taylor (1948, p.457) which were obtained using the friction circle method. The dashed lines are values of N from the Culmann solution, Eqn (5.3). For slopes close to vertical the Culmann solution agrees quite closely with the results of the friction circle method, however the two solutions diverge for lower values of 6  The friction circle method agrees closely with results obtained from the ordinary method of slices 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 widely 111  used today. However the stability curves developed by Taylor (1948) can be applied to some special 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 particular  soil values y c’ and ‘. Alternatively stability curves of  versus 9 can be constructed for  known values of y,, and c’ as in Fig 5.3(b).  To develop stability curves for an applied river bank analysis, field investigations are necessary to determine the bank stratigraphy, the properties of the bank sediments, and the likely shape of the failure surface. Also other variables such as the presence of surface tension cracks and the influence of bank vegetation on the stability of the bank can be considered. Groundwater monitoring 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 the curves can be approximated by empirical equations or piece-wise linearized segments, and then this information can be accessed quickly by the optimization model. This is necessary as the optimization model must assess the stability of hundreds or thousands of H-9 combinations before arriving at the optimum solution.  5.2.1.1 Bank Stability and Submergence The effect of submergence and emergence of the channel banks due to flow variability plays a key role in the stability of the channel banks. Bank submergence results in increased soil weight due to saturation, and a decrease in the friction resistance due to increased pore water pressures. These two effects may combine to reduce the stability of the banks. This decrease in bank stability may however be countered by increased hydrostatic loading along the bank surface.  Three limiting cases will now be considered that can be analysed using the stability curves  developed by Taylor (1948). These cases are shown in Fig 5.4 and will be used to demonstrate 112  the effect of submergence, saturation, and emergence on the stability of a river bank. The values of c’ and  ‘  to be used in the following discussion are assumed to equal the developed  values, that is the factor of safety applied to the values c’ and qS’ equals 1, and the values c’ and are assumed to be known exactly.  Case I corresponds to a condition of zero or low flow in the channel, together with drained banks. The value of N 3 can be determined directly from the stability curves for a particular value 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 is being supported by buoyancy forces about its perimeter as indicated in Fig 5.3. For this case when the soil is fully saturated, and there is no seepage, the buoyancy forces are hydrostatic. 3 can be determined directly from the stability curves, and The value of N using the buoyant unit weight,  Yb’  then calculated  which is equal to saturated Yt minus y.  Case III corresponds to a condition following complete and instantaneous drawdown of the flow in the channel. In this case the soil in the bank remains saturated, the pore pressures unchanged from the fully submerged case. However the hydrostatic forces which were previously acting to stabilise the bank are now absent. Taylor (1948) determined that the value of N for case III could be approximated by modifying the value of ‘to account for the pore pressures acting along the failure surface which reduce the frictional resistance. It was S,,, can be approximated determined by Taylor (1948, p. 467) that the modified friction angle 9 by: (5.6)  Where y 1 is the saturated value. This relation, which applies only to complete and instantaneous drawdown, has been confirmed by Morgenstern (1963) who developed stability curves for 113  complete and partial instantaneous drawdown using the Bishop simplified method of slices. For Case III the value of N., is determined from the stability curve corresponding to  qSm,  and  calculated using y . 1  The values of N., and 11 cr11 will now be determined using reasonable soil values to demonstrate the relative stability of the three cases. A value of q’ , saturated y 3 kN/m  =  22.76 kNIm , and Yb 3  values give a value of  Øm  =  25° together with the drained Yt = 20.80  12.95 kN/m 3 for 0=  =  700.  From Eqn (5.6) these  14.2°. The values for ]V are read from the stability curve from  =  Taylor (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 Yr qY c’  OT Yb  ) 3 (kN/m (°) ) 2 (kN/m  M, (m)  The values of  Case I  Case II  Case III  20.80  12.95  22.76  25.0  25.0  14.2  10.0  10.0  10.0  9.8  9.8  7.2  4.71  7.56  3.16  in Table 5.1 indicate the role of submergence and emergence on the stability  of the channel banks. Note from Case II that the effect of increasing the flow depth is to stabilise the channel banks and increase  Therefore the limiting condition for bank stability  must be satisfied at low or zero flows. Cases I and ifi defined upper and lower limits for the bank stability. The critical period for bank stability is during flow recession. Case I corresponds to a river with very slow runoff response and/or very permeable bank sediment. As the flow recedes the banks must be dewatering so that the bank above the water surface in the channel is fhlly drained, and the bank below the water surface fi.illy saturated. This continues until zero 114  flow condition and the banks are totally drained (Case I). The other limiting condition corresponds to a river with flashy runoff and/or highly impermeable channel banks. The flow recedes at a rate at which no dewatering of the bank sediment occurs and the banks remain fl.illy saturated at the zero flow condition (Case III). The stability of natural rivers would lie between the two limiting cases (I and III).  5.2.1 2 Bank Height Constraint The bank height constraint requires that the bank height H, be less than or equal to the critical bank height  HH  where H is equal to the bankfull flow depth 1’, the value for N obtained from the stability curve. The value of  (5.7)  is calculated by Eqn (5.2) with  should be evaluated for both Case I and  Case III conditions. Examples of stability curves derived from field studies can be found in March 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 Erosion Fluvial erosion is the process whereby individual grains or aggregates on the surface of the bank are removed and entrained by the flow. The driving force is the shearing action of the fluid on the bank sediment. The mechanics of fluvial erosion of cohesive sediment remain poorly understood despite considerable research. Reviews of the research conducted on the erodibility 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 of several forces of attraction and repulsion. These forces result from very complex electro chemical processes which include the clay mineralogy and content, and the temperature and  chemistry of the pore and eroding fluids (Arulanandan et a!., 1980; Gressinger, 1982; 115  Raudkivi, 1990). Unlike noncohesive sediment, the weight component of the cohesive grains is quite insignificant when compared to the electro-chemical forces. The electro-chemical forces may 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 performed using a variety of techniques such as straight and circular flumes, rotating cylinders, submerged jets, 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 critical shear stress, critical velocity, or critical stream power which must be exceeded before erosion commences.  Attempts have been made to use the results from these erosion studies to develop correlations between the erodibility of the cohesive sediment and other primary physical properties of the soil which are more readily obtained. These physical properties include clay content, clay chemistry 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. In general this approach has not been overly successful and has not yielded results which could be readily used for predictive purposes. However the approach of Arulanandan et al. (1980) has produced charts which relate soil erodibility to the electrical conductivity and the magnitude of the dielectric dispersion of the soil, properties which reflect the type and amount of clay. It addition it was shown that the chemistry of the eroding fluid, expressed as the sodium adsorption ratio, also significantly influences the soil erodibility. According to Arulanandan et al. (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”. Note that the same soil can have different values of r for different eroding fluids, and therefore the value of  corresponds to a particular soil-water system.  116  The critical shear stress concept will be adopted in this thesis. The critical shear stress of a particular soil ;,,, can be evaluated directly using one of the erosion devices mentioned previously, preferably on undisturbed in situ samples, or it can be estimated indirectly by methods such as Arulanandan eta!. (1980).  5.2.2.1 Bank Shear Constraint The bank shear constraint requires that the bank shear stress be less than the critical value required for fiuvial erosion of the bank sediment:  ;aflk  where the quantity  Vbank  (5.8)  is calculated from Eqn (4.8). This constraint assumes that the gravity  component 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 independent property of the soil-water system. The assumption that a soil water system has a single definable value of  belies the  complexity of the erosion process. For example freeze-thaw action, ice wedging, and desiccation and tension cracks can all result in aggregates of cohesive soil which may have a significant gravity component and a much lower value of r than a smooth homogeneous soil mass. Nonetheless for modeling expedience it will be assumed that a soil-water system can be characterised by a single value of z which is treated as an independent variable.  5.3 NONCOHESIVE BANK SEDIMENT As with cohesive sediments both mass failure and fiuvial erosion will be considered for noncohesive sediment.  117  5.3.1 Mass Failure For a bank composed of noncohesive sediment the resisting force that balances the driving force is due only to the friction term in Eqn (5.1). It is an elementary exercise to show that the factor of safety for an infinite, drained slope of noncohesive sediment is given by:  F  =  tanq5 tan8  (5.9)  from which the limiting stability occurs when the bank angle 8 is equal to the angle of internal friction  .  Therefore for a bank to be stable the following constraint must be satisfied:  8qS  If 8 exceeds  (5.10)  0, bank failure will result.  When a bank is fully submerged as in Case II from the previous section, the stability is unchanged from the fully drained case as the buoyancy forces reduce the driving force and the resisting force by the same amount. For a condition of rapid drawdown seepage forces can develop which reduces the stability of a noncohesive bank. Since only frictional resisting forces are involved, the decrease in bank stability due to seepage forces can be simulated numerically by 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)  ‘Ft  In this case the value of  is analogous that obtained by Taylor (1948) for cohesive sediment  (Eqn 5.6).  118  However in many banks composed of noncohesive sediment, particularly those formed of coarse gravel, significant seepage forces will not develop due to their high permeability. The river 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 Erosion In flowing water the fluid forces exerted on the grains as well as the down slope gravity component 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 United States Bureau of Reclamation (USBR) to determine the stability of noncohesive bank sediment. Its development is summarised in Lane (1955b) and it is presented in the following simplified form from Chow (1959, p.171) and Henderson (1966, p.419) for the limiting bank stability: 2  Tbaflkc  —  = Tbedc  where  rbaflkc  J  6 Ø 2 sin  (5.12)  is the critical shear stress for a grain located on a sloping bank, and  Vbed  is the  critical 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:  I  Vbank  ( _in r\s  S1fl6  sin 2  (5.13)  where s is the specific gravity of the sediment, D5Obaflk is the median bank grain diameter which is assumed to be representative of the bank sediment,  bed  is the critical dimensionless  on the channel bed. (Shields) shear stress for a grain equivalent to D50 bank 119  Unlike 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 to account for the properties of the bank sediment.  USBR data (in Lane, 1955b) indicate that qS for coarse gravel approaches a maximum of approximately  400.  However consolidation and imbrication of the bank sediments, together  with the effect of cohesive silt and clay cementing the grains and other stabilising influences can increase the in situ friction angle above  0.  Therefore çS in Eqn (5.13) can be replaced by the  in situ friction angle,  Ø, which can be allowed to take a maximum value up to 900.  The value of  must also be adjusted because the critical shear stress for the stabilised  t*bedc  bank 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 Chapter 6.  To include the influence of bank vegetation, consolidation, and cementation of the bank sediments,  0  is replaced with  Substituting Eqn (5.14) into (5.13), and using  ,  the  resulting constraint for noncohesive banks is:  bank  ll— 1 ktanqS  120  Sifl  sin  6  (5.15)  Note that Eqn (5.15) is presented as an inequality constraint with the mean bank shear stress r bank’  rather than  ;ank  in the numerator on the left hand side. The value 6,, can take a maximum  value 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-root term becomes undefined for values of 9>. Eqn (5.15) therefore simultaneously constrains the bank stability with respect to both mass failure and fluvial erosion, unlike the noncohesive banks where two separate constraints are required.  The influence of the bank stability on the channel geometry will be demonstrated in Chapter 6.  5.4 SUMMARY The bank stability constraints were developed for both cohesive and noncohesive bank  sediment. For both sediment types there are two mechanisms of bank erosion that must be considered, namely mass failure and fluvial erosion of individual particles.  For cohesive sediment the stability of the bank with respect to mass failure can be assessed using slope stability techniques. To incorporate the slope stability data into the optimization model stability curves are constructed. These curves can be accessed rapidly by the optimization model. The banks are most susceptible to failure at low or zero flows when the hydrostatic supporting force from the flow is absent. Two limiting cases are recognised, fully drained, and fully saturated bank conditions. For bank stability the cohesive bank sediment must also be stable with respect to fluvial erosion. The concept of critical shear stress for cohesive sediment is used. The bank shear constraint requires that the value of 121  Tbank  be less  than or equal to the critical shear stress for the bank sediment. The value  ;ank  represents the  mean bank shear stress. In practice it is the maximum bank shear stress that would likely occur near the toe of the bank, rather than to fluvial erosion. Nonetheless  ;ank  ank  that would determine the bank stability with respect  will be used.  To assess the bank stability of noncohesive bank sediment the USBR bank stability presented by Lane (1955b) will be used in a modified form. The original equation is modified to reflect the 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 in situ friction angle  ,  that can take a value up to  900.  The stability of noncohesive bank  sediment with respect to both mass failure and fluvial erosion is assessed by the single USBR relation except in the case of large seepage forces following rapid drawdown. The influence of large seepage pressures in noncohesive sediment will not be considered as the high hydraulic conductivity of these sediments, particularly for coarse gravel sediment, would facilitate rapid draining of the bank.  Gravel-bed rivers commonly have channel banks that are composite in nature and typically have a lower noncohesive unit which is overlain by an upper cohesive unit. In the case of composite 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 layer then undergoes mass failure as it becomes undermined (Thorne, 1982). In this example it appears 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.  122  H  FD  Figure 5.1. Bank stability analysis for planar failure surface.  123  =  wsina  =  cL + NRtan  (a)  H  L  (b)  Figure 5.2. Bank stability analysis for the method of slices. (a) Definition sketch. (b) Forces resolved on a single slice. The lateral forces not indicated.  124  25  (a)  2O a,  .0  E  z 15 .0  1o  5 8(i)  (b)  10 8 6  4  2  20  40  60  e ()  80  Figure 5.3. Stability curves used to determine Hrit. (a) Stability number as a function of Critical height as a function of Ofor prescribed values of c’ and rt.  125  (b)  Case I  Case II  A  Case III  Pore Water Pressures Pressures  Figure 5.4. Three special cases for bank stability analysis. (a) Case I: Drained banks. (b) Case II: Fully submerged and saturated. (c) Case ifi: Instantaneous and complete drawdown, bank fi.illy saturated. The hydrostatic and pore water pressures are indicated.  126  CHAPTER 6  EFFECT OF BANK STABILITY ON CHANNEL GEOMETRY  6.1 INTRODUCTION In Chapter 6 simplified versions of the optimization model will be presented whereby the  sediment 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 bedload transport relation will be used. This simplified model is inadequate to assess the response of a natural channel to variations in the discharge or sediment supply, but is sufficient to investigate the influence of the bank stability on the channel geometry, and to introduce the optimization methodology to be fully developed in Chapter 7.  6.2 BANKFULL : fiXED-CIIANNEL-SLOPE OPTIMIZATION MODEL  In the bankfiill model the natural variation in the flows is reduced to a single dominant or channel-forming discharge which is assumed to be equal to the bankfull discharge, Qbf The sediment transporting capacity of the channel is indexed by the sediment transport rate corresponding to the bankfull discharge, Gbf This simplification has been used by Chang  127  p  (1980), White et a!. (1982), Millar (1991), and Millar and Quick (1993 a, b) in their optimization analyses of gravel rivers, and Hey and Thorne (1986) in their empirical regime analysis. The limitations of this approach with respect to calculating the sediment transporting capacity of the channel were discussed in Chapter 4. The bankfiull model is however adequate to 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 the first version a further simplification will be made in that the channel slope S will be treated as an independent variable. This is done to assess the effect of the bank stability on the values of W and  Y  without the interference of varying channel slope. In Section 6.5 the model will be  generalised to include variable S.  The bankfull:fixed-channel-slope model is analogous to an experimental setup where the slope is fixed, and the channel width, depth, and sediment transport rate adjust to the imposed discharge (eg Wolman and Brush, 1961). The model is formulated below.  6.2.1. Independent Variables  The variables that are assumed to be independent with known values are Qb S, , 50 D d , 5 0 k3bj, the bank stability variables which are D5Oba and ç for noncohesive banks, and c’, and  ‘,  y,  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 in this formulation.  128  6.2.2. Dependent Variables The primary dependent variables to solve for are  and 6 From these primary  Pbed, Pbank,  variables the secondary dependent variables such as j U  Rh,  W, Y,  gb,  and others can be  determined.  6.2.3. Objective function The objective function in this formulation is given by:  (6.la)  maxf(,1,6)=77 where:  (61b) pQbfS  As the values of p, Qb, and S are prescribed, maximization of  i  is equivalent to the  maximization of the product Pbed times gb. The value of gb will be calculated using the EinsteinBrown sediment transport relation as presented in Vanoni (1975, p. 170). This relation requires 50 to calculate gb: only the value of d  *  gb  12.15 exp(_0.391 /  L4o  <0.093 (6.2a) r  50  50  0.093  where gb*= dimensionless bedload transport rate per unit width given by:  129  *  gb=  gb 50 1 ps,.J(s—1)gd  The dimensionless bed shear stress for the median bedload-grain diameter  *  ‘Cd  is given by:  Tbed  —  62 C  r(sl)d  The value of ‘Cbed is calculated from Eqn (4.9).  0 and is given by: The parameter F 1 is related to the fall velocity of the sediment d,  12  36v 2 3(s—1) 50 1j3gd  I  36v 2 3(s—1) 50 1Jgd  (6.2d)  1 is approximately equal to where v = kinematic viscosity of water. For gravel-sized sediment, F 0.82.  6.2.4 Constraints The only two constraints that are required in this formulation for uniform flow in a prismatic channel are continuity and bank stability. As the value of S is prescribed the bedload constraint is not required.  130  6.2.4.1 Continuity The continuity constraint determines the dimensions of a channel which can just convey the imposed  Qbf  The continuity constraint for uniform floe in a prismatic channel is defined as:  (3.24)  UA=Qbf  where U is the mean velocity, and A is the cross-sectional area of the flow. The value of U is obtained from the Darcy-Weisbach equation:  I8gRS  (3.25)  U=f  The value of the friction factorfis calculated using Eqn (3.19).  6.2.4.2 Bank Stability Constraint The single bank stability constraint for noncohesive bank sediment is given by:  I  Tbank  SiflO  , ktanq1_Sfl2 501 r(s—l)D  131  (5.15)  where the value of  Thank  is given by Eqn (4.8). The value of the constant k is to be evaluated  from field data in Section 6.3.1.  For cohesive bank sediments the bank stability constraints consists of the bank-height and the bank-shear components:  HHmt  (5.7)  r,  (5.8)  Tbank  Both constraints must be satisfied for cohesive banks to be stable.  6.2.5. Optimization Scheme: Fixed-Channel-Slope The optimization flowchart is shown in Fig 6.1 and the source code for the computer program is given in Appendix A. The program is a step-wise iterative procedure that varies only one  dependent variable at each stage. The program is initialised by selecting trial values of the three primary dependent variables, value Qbj  Of P bank  for trial values  Pbed, Pbank,  0. The continuity constraint is satisfied by varying the  and  When the continuity constraint has been satisfied for  Of Pbed  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 stability  132  constraint, however these lower values do not represent the optimum bank angle. Fig 6.2 shows the typical variation of  Tj  and  r,ank  for a range of values 6 for which the continuity  constraint has been satisfied. Note that the value of  rd  decreases monotonically with  decreasing 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 efficient  with respect to transporting sediment than a channel with a bank angle equal to O,,,. The final optimal solution must have a value of 8 that corresponds to a value of O,,,. Trial values of 8 are assessed until  m 6  is determined. For each trial value of 6, a new value of P bank must also  be determined which satisfies the continuity constraint.  Once the value of 8,,, has been determined, the sediment transport efficiency variation of  i  over a range  Of Pled  i  is assessed. The  values for a prescribed value of S is shown in Fig 6.3. Each  point on the solution curves in Fig 6.3 satisfies the continuity and bank stability constraints. The  value of  i  is reduced to zero either when Pbed equals zero, or when Pbed is very large and Y  becomes very small, and the value of r,,ed, and hence gb, both approach zero. The optimal value of 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 and locating the optimum. This method requires that the initial upper and lower bounds of the search be initially specified for the variable in question. The midpoint between these two bounds is evaluated and, depending upon the outcome, becomes the upper or lower bound for the next stage of the search, thus reducing the  search area by half For example to satisfy the 133  continuity constraint upper and lower values of Pbank are established which are sure to contain the final value of Pbank. The midpoint between these two values is evaluated. If the calculated discharge capacity of the channel at the midpoint is greater than Qj the midpoint Pbank value is too large, and this value then becomes the upper bound for the next stage. Conversely if the discharge capacity is less than  1 Qb  the midpoint value of P bank then becomes the lower bound  for the next stage. Convergence is obtained when the calculated discharge capacity falls within a selected tolerance of Qb 1 say ± 0.1%.  7 / dPbed A similar procedure is used to evaluate the optimal value of Pbed. The first derivative dr is evaluated numerically by cental differencing at the midpoint between the upper and lower 7 / dPd> 0 the midpoint becomes the lower bound, and for bounds to Pbed. For values of di 7 / dPd <0 the midpoint becomes the upper bound for the next stage. Convergence values of di is attained when the separation between the upper and lower bounds of the search is reduced below 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 Newton Rapson method were generally more computationally efficient, however they were prone to instability and occasionally caused the program to crash by returning negative values for the dependent variables.  This iterative search scheme assumes that the functions are well behaved with no local optima.  134  This is a necessary requirement for this type of iterative search procedure, as well as other nonlinear optimization techniques such as the reduced gradient method, which can get caught up in local optima. At a local optimum all of the convergence criteria can be satisfied and yet it not possible to determine whether a local or global optimum has been found. Different optimal solutions can be obtained from different starting points for the search procedure. The variation of the value of the objective function  i  with Pbed for the fixed slope bankfull model is shown in  Fig 6.3. The objective function is smooth with only a single global optimum. Evidence of local optimum 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 small  for low values Of Pbed because the high shear stresses acting on the banks. As Pbed increases, the value of Tbank decreases and banks with higher values of 6 are stable. In the example used in Fig 6.3 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 with noncohesive and cohesive bank sediment.  6.3 NONCOHESWE BANK SEDIMENT  The analysis to be presented in this section is similar to that described in detail in Millar and Quick (1 993b), however there are significant differences which result in differences between the numerical values obtained herein, and those from Millar and Quick. The general conclusions are however unchanged.  135  The model will be tested on the published gravel river data collected by Andrews (1984) and Hey and Thorne (1986). The published data will be used as input to the model, and the output geometry will be compared to the observed geometry. The relevant data and modeffing results are tabulated in Appendix D.  These rivers are described as stable single-thread channels with mobile beds that actively transport sediment at higher stages. The banks are either comprised of noncohesive gravel similar to the material being transported, or have composite banks with a lower noncohesive unit, overlain by a cohesive silty unit. For the composite banks it will be assumed in this analysis that the stability of the banks is determined by the lower noncohesive unit. The banks are characterised in terms of their vegetation density. Andrews (1983) subdivided his data set into those channels with thin and thick vegetation. Similarly, Hey and Thorne (1986) subdivided their data into four bank Vegetation Types ranging from Vegetation Type I, grass with no trees or bushes, to Vegetation Type IV, with> 50% trees and bushes.  The empirical regime analyses of Andrews (1984) and Hey and Thome (1986) determined that the effect of the bank vegetation was to increase the stability of the channel banks which resulted in narrower channels for the same value of Qbf In the first stage of the present analysis only the channels which have low densities of bank vegetation will be assessed. It will be assumed that the bank stability of these channels is determined by the bank sediment properties alone, and is unaffected by the bank vegetation. This preliminary analysis will permit the  136  estimation of the bank stability parameters. In subsequent analyses (Section 6.3.2 and 6.3.3) the channels with higher densities of bank vegetation will be examined and the effects of increased bank stability on the channel geometry due to the bank vegetation will be demonstrated.  50 The values of several key parameters were not explicitly given in the data sets. The value of d 0 I 3 will be used. This D was not given by Hey and Thorne, and as an approximation 5 approximation will affect the absolute value of g calculated for the channel, but will not influence the location of the optimum significantly.  Neither Andrews nor Hey and Thorne specified the values of D50k. In this analysis the value Of D5Oba,  50 the median grain diameter of the armour layer sediment. D will be assumed to equal ,  Banks composed of sediment similar in size to the transported bedload sediment would be 0 the median bedload or subarmour grain , 5 expected to have a value of D5Oba,,k similar to d diameter. However previous work (Millar, 1991; Millar and Quick, 1993b) has shown that very poor results were obtained using D5ob D5Qba  =  =  0 and much better results were obtained using , 5 d  0 This is consistent with the development of a coarse static armour layer on the . 5 D  channel 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 toe of the bank.  ,g = d 0 50 will be shown to be valid in Section 6.3.1. The assumption ofD5o  137  The relative roughness values of the bed and banks are not known. Therefore since the bed and banks are composed of similar sized gravel, the values of ksd and ks,flk will be assumed to be the same, and both equal to the value of k which was calculated from the observed channel geometry by inverting Eqn (3.2).  6.3.1 Low Density Bank Vegetation The channels which will be analysed in this section are the channels described as having thin bank 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 to the critical dimensionless shear stress bank sediment  .  V*be4  by Eqn (5.14), and the insitu friction angle of the  A value of çS, = 400 will be used as input which assumes that the banks are  comprised of loose, noncohesive gravel. A wide range of values for  VCbe  are cited in the  literature, with most faffing between 0.03 (Neil, 1968) and 0.06 (Egiazaroff 1965). For a value of ç  =  400  this translates into a range of values for k between 0.036 and 0.072. The value of k  within this range that gives the best agreement between the observed and modelled channel widths for the 27 channels will be determined, and then assumed to be a constant for all channels.  The model was run for a range of values of k to determine the best fit between the observed  and modelled channel widths. The optimum value was determined to be k = 0.048 which is  138  equivalent 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 rather than D5obQ,,k  50 D ,  50 Typically in gravel-bed rivers the value of 1),o of the armour layer is d .  approximately three times d 50 from the subarmour sediment. If the assumption that Dsob is valid, then the value of k required to force an agreement between Wobs and  50 d  would be  approximately three times as large than the value that is obtained when using D, 50 This D 0=. value of k would correspond to a value of  equal to about 0.12, which is well outside the  range of values usually reported in the literature.  Comparisons between the modelled and observed channel widths and mean depths are shown in Fig 6.4(a, b). The agreement is good, the mean value of W b I Wmod is 1.00 together with a 0 coefficient of variation of 28.4%, and the mean value of Y b / Ymod is 1.05, and the coefficient 0 of variation is equal to 14.5%.  6.3.2 Effect of bank vegetation The model was run with  =  40° for the remaining 58 channels; those described as having  thick bank vegetation by Andrews, and the Vegetation Type II  -  IV channels from Hey and  Thorne. Reach 9074800 from Andrews’ data set was excluded from the analysis because a stable channel width could not be obtained with çz5 = 40°.  The results including the channels with the  low density of bank vegetation are shown  139  graphically in Fig 6.5(a, b). Considerable scatter is evident away from the line of perfect agreement. Note the strong asymmetry in the scatter. The modelled channels are consistently wider and shallower than the observed channels. The tendency for the modelled channels to be wider and shallower than the observed channels increases with the density of the bank vegetation. The channels with thin bank vegetation and those classified as Vegetation Type I are scattered symmetrically about the line of perfect agreement as they were forced to do by the selection of the value for k in the previous section. The channels classified as having thick bank vegetation, and the Vegetation Type IV channels tend to lie furthest from the line of perfect agreement.  The values of the ratios W b / Wmod and Yobs / 0  Ymod  for each Vegetation Type are summarised  in Table 6.1. Despite the wide scatter within each group it is evident that these two ratios show d decreasing, and 10 b / W,, 0 a systematic variation with the density of the bank vegetation with W 3/ b 0 Y  Ymod  increasing with increasing bank vegetation density.  Table 6.1. Ratios between the observed and modelled values of W and Y for fixed-slope optimization model with q = 400. Vegetation Type I II III IV Thin Thick  W,b /  Mm 0.64 0.48 0.28 0.25  0.74 0.17  W Mean Max 1.77 0.97 1.34 0.83 1.16 0.72 1.00 0.59 1.44 1.03 0.82 0.58  I  Y / Y, Mean Mm 1.05 0.70 1.13 0.83 1.28 0.90 1.48 0.98 0.74 0.99 1.31 1.11  140  ,  Max 1.29 1.50 2.05 2.16 1.16 1.44  No. of Rivers 13 16 13 20 14 9  The results obtained from the optimization modelling will now be compared to the those results obtained by Andrews (1984) and Hey and Thorne (1986) using empirical regime analyses. These researchers used regression analyses to derive empirical regime equations for the hydraulic geometry. Separate equations were obtained for each bank vegetation density. The width equations were of the form:  W=aQb/  where  Qbf  (6.3)  is the bankfull discharge in Hey and Thorne, and is the dimensionless bankfull  discharge in Andrews. The exponent b shows little variation and typically takes a value of approximately 0.5. The coefficient a was found to vary with the density of the bank vegetation becoming smaller as the bank vegetation density increased.  The ratios formed by dividing the coefficient a from each of the regime equations, by the coefficient a from the equation which represents the channels with the lowest density of bank vegetation is equal to W 3 / Wunveg. where the subscript b 0  unveg  refers to the unvegetated channel  width. The ratio W b / Wunveg is an index of the influence of the bank vegetation on the channel 0 width. For example the value of a for the Vegetation Type I channels of Hey and Thorne is 4.33, and the value of a for the heavily vegetated Type IV channels is 2.34. Therefore the Vegetation Type IV channels are on average 2.34 / 4.33 = 0.54 times as wide as the weakly vegetated Type I channels.  141  The ratio W 3 / Wmod where Wmod is b 0 b / Wunveg is directly analogous to the preceding ratio W 0 the calculated width assuming qS =  400.  The values of W b / Wunveg for the Andrews’ data set 0  and the Hey and Thorne data set are summarised in Table 6.2 together with the values for W b 0  I Wmoj obtained from the modelling in this thesis with  = 40°. Clearly there is reasonably good  agreement between the results of the optimization modelling with  = 40°, and the empirical  regime analyses of Andrews and Hey and Thorne.  Table 6.2. Summary of ratios of observed channel width divided by the unvegetated channel width. These ratios are an index of the effect of bank vegetation on the channel width. The values from Thorne et a!. (1988) were calculated using their four regression equations and an assumed value of Wmod equal to 25m. See text for discussion. Vegetation Type  b / Wmod 0 W This Thesis  I II III IV Thin Thick  0.97 0.83 0.72 0.59 1.03 0.58  b / Wunveg 0 W Andrews (1984) -  -  -  -  1.00 0.79  Wob / Wunveg Hey and Thorne (1986) 1.00 0.77 0.63 0.54  Wobs / Wmod Thome et a!. (1988) 1.45 1.20 0.96 0.84  -  -  -  -  Also in Table 6.2 are results from Thorne et a!. (1988) who tested the optimization model of Chang (1980) on the data set of Hey and Thorne (1986). The essential difference between the optimization model developed by Chang, and the fixed-slope model from this chapter, is that Chang assumes a constant value for the bank angle Ofor a given channel, and does not consider bank stability.  142  A direct comparison between the results obtained from Chang’s optimization model and the results obtained in this thesis is not possible as the values of Wmod were not given by Thorne et al. However an indirect comparison is possible. To account for the influence of the bank vegetation Thorne ci al. developed four regression equations, one corresponding to each Vegetation Type. This allowed Wmod obtained from Chang’s model to be corrected for the influence of the bank vegetation. Although the values of Wmo€i from Chang’s model are not known, the four regression equations can be used to back calculate a value of Wmod for each  b / Wmod for a selected value of 0 Vegetation Type for a selected value of W . The values of W 3 b 0 Wmod  =  25 m are listed in Table 6.2.  The results from the Thorne et al. analysis using Chang’s optimization model are significantly different from the results of Hey and Thorne and in this thesis. This demonstrates that the inclusion of the bank stability constraint makes a very significant improvement in the performance of the optimization model.  6.3.3 Influence of ç6 on Channel Geometry  The preceding section indicates that the bank vegetation has a large influence on the bank stability, which in turn has a correspondingly large influence on the channel geometry. The effect 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 the grains by the root masses. A simple method of accounting for this effect is by modifying the  143  value of  which would take larger values for channels more strongly affected by the bank  vegetation. This approach ignores the cohesive effects that would be introduced by the root masses, and is simply a device to account for the influence of the roots that can be incorporated into the noncohesive bank-stability constraint (Eqn 5.15).  To demonstrate this effect the optimization model was programmed to run for a series of trial values of ç& for each of the channels from Andrews (1984) and Hey and Thorne (1986) until the value of Wmod was equal to W b within a tolerance of ±1%. For example, Reach 13 from 0 Hey and Thome has a value of Wobs 400  a value of Wmod  =  =  18.4 m; when the optimization model was run with q 5  74.4 m was obtained, which is over four times the observed value. After  successive trials it was determined that a value of Wmod  =  =  73.10  forced an agreement between  and Wobs.  The values of  Ø  obtained from this analysis depend on the value of D 50 assumed. Because  the bank stability constraint (Eqn 5.15) is a function of two parameters, qS,. and D 50 (the  values of k, s, and yare assumed constant), the value of  can only be determined if D5obO is  50 Using different values of D D known, or in this case is assumed to equal . 50 will result in different 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 program  144  result 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 large scatter within each vegetation grouping there is an observed increase in the mean values of q5 with increasing bank vegetation density. This supports the general approach of accounting for the effect of the bank vegetation by adjusting  .  The optimization model was rerun for all of the channels using the mean values of g from Table 6.3 for each vegetation subdivision. The results are shown in Fig 6.6(a, b). There is a reduction in the scatter when compared to the results shown in Fig 6.5(a, b) for the mean values of /  Wmod  =  40°. Using  for each Vegetation Type, for the complete set of channels the ratio W b 0  has a mean value of 1.00 and a coefficient of variation of 28.4%, and  Yobs  / Ymod has a  mean 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) and Hey and Thorne (1986). Vegetation Type I II III IV Thin Thick  No. of Rivers 13 16 13 20 14 9  Mm (°) 21.1 31.5 34.2 39.7 30.4 45.7  çS values Mean (°) 42.0 47.0 53.0 60.1 40.2 55.7  Max (° 52.8 61.0 72.5  90.0 48.4 67.6  The residual scatter in Fig 6.6(a, b) can be attributed to several reasons. These include:  145  1. The bank vegetation categories are subjective, and a continuous gradation of the vegetation density exists within and between the vegetation types, and therefore a corresponding gradation in q5,. would be expected.  2. The effect of the bank vegetation is not simply a function of the vegetation density, but is undoubtably dependent upon vegetation type, age, and rooting depth, as well as the thickness of the overlying cohesive unit.  3. Imbrication, packing, and cementing of the gravel by fine sediment is independent of the Vegetation Type.  4. The assumption that D5Obaflk  =  50 is an additional source of uncertainty. D  6.4 COHESIVE BANK SEDIMENT The model will now be formulated for cohesive bank sediments and tested on hypothetical and real river data. The only change from the formulation for cohesive sediments is the bank stability constraint.  As was discussed in Section 5.2 the bank stability constraint for cohesive sediment is composed of two individual constraints, namely the bank-height and the bank-shear constraints which correspond to the erosion mechanisms of mass failure and fluvial erosion respectively.  146  6.4.1 Bank Stability Routine  The bank stability routine satisfies the bank-height and bank-shear constraints consecutively. First the bank-height constraint is satisfied. The values of H and Hrrt are initially assessed for 8 =  900.  For 6= 90°, and for subsequent values of 8, the stability number N , is obtained from 3  stability 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 the continuity 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 not satisfied, the value of 8 is reduced until H = Hcrjt which gives the maximum bank angle 8 for which the bank-height constraint is just satisfied. For values of 9  <  8,,, the bank-height  constraint is satisfied, but the channel will be less efficient with respect to transporting sediment as was discussed in Section 6.2.5.  Unlike the noncohesive case, the stability of cohesive banks with respect to fluvial entrainment  does not necessarily increase with a reduction of 8. Fig 6.2 shows the typical variation of  Tbank  with 8 where continuity has been satisfied. As the value of 8 is reduced from the maximum  value,  Tbank  increases to a maximum, and then decreases for low values of 9. Therefore except  for small values of 8, a reduction in 8 is accompanied by an increase in  , 07 Tb  and a reduction in  the stability of cohesive banks with respect to fluvial erosion. The bank stability routine assesses the bank-shear constraint for the value and if Tb J 0  which satisfies the bank-height constraint,  r then the bank-stability constraint has been satisfied. If  value of 8 is reduced until  Thank  =  T.  In  rit for 8,,,, the  practice it has been found that if the bank147  shear constraint is not satisfied for about  200  then the value of 8 which satisfies the constraint is  or less.  The full data requirements for model testing were not available, however the model will be tested on the published data from Chariton et a!. (1978) which lists many of the required input values.  6.4.2 Data From Chariton et aL (1978)  Fourteen rivers from the Chariton et a!. (1978) data set which were described as having banks composed of fine sediment will be used. The rivers, together with the hydraulic geometry and other relevant information are listed in Appendix E. The value of d 50 for the subarmour sediment which is used to calculate gb was not given in the data set, and as with the noncohesive channels in Section 6.3, the value d 50 = D 50 / 3 will be used as an approximation.  The value of ksbed was set equal to k 5 calculated from the observed channel geometry. Unlike gravel-bed rivers with noncohesive gravel banks, those with cohesive banks may have significantly different values of  and ksbank.The value of  is unknown, so a value equal  to 0.1 m was assumed for all channels. This assumption will not significantly affect the result as the total channel roughness is determined largely by ksbed•  The value of the unconfined compressive strength q was given for each channel. This value is an average of several samples from each river. The friction angle for the bank sediment was not 148  given and a value g’  =  25° is assumed for all channels which is a reasonable value for the  moderately 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 assumed for all channels for drained bank conditions, and  rt  20.0 kNIm 3 will be  22.5 kN/m 3 for saturated bank  conditions. For the saturated bank conditions the value of ‘ will be modified by Eqn (5.6) to reflect 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 of N will be calculated from the stability curves in Fig 5.3(a). These stability curves were approximated by piece-wise linearized segments over the range  0’  6  90°. Although  these curves were developed for homogeneous, drained soils which are unlikely to exist in actual field situations, these curves will be used here for illustrative purposes. More realistic stability 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 directly estimate the values for z. However in a manner similar to the estimation technique for  149  for  the noncohesive channels in Section 6.3.3, the optimization model can be used to indirectly estimate values for  Because the value of T rif is not known, initially the model will be run with a very large value for Tcrjt (crit =  1000 N/rn ) to ensure that the bank-shear constraint is not influencing the solution 2  and the bank-height constraint alone is analysed. From this analysis the channels that are potentially bank-height constrained can be determined. For the remaining bank-shear constrained channels the values of rrjt can be estimated by forcing an agreement between Wmod and W 5 as was done for b 0  Ø in Section 6.3.3 for the channels with noncohesive banks.  The values obtained from the model assuming  rrgt =  1000 N/rn 2 are presented in Table 6.4 and  Fig 6.7(a, b). The model was run for both drained and saturated bank conditions. From the values of H / Hcrjt it is evident that channels 5, 7, 10, 12, and 13 (and 8 for saturated bank conditions) are constrained by the bank height. The remaining channels are bank-height degenerate. (The term degenerate is used in optimization to denote constraints that are not actively constraining the solution.) The bank-shear constraint is degenerate for all of the 14 channels due to the large imposed value of Thrit. The bank-height constrained channels will now be considered. Note that different solutions are obtained for the drained and saturated bank conditions. The saturated banks are inherently less stable than the drained banks due to the increased unit weight of the soil, and the increased pore pressures. This effect is well known  and is discussed in Section 5.2.1.1. The second feature evident from this analysis is that the bank-height constrained channels tend to be the larger channels with values of Qbf greater than  150  5 7 8 10 12 13 1 2 3 4 6 9 11 14  Reach No.  W (m) 17.6 31.0 28.7 19.0 39.3 59.4 17.4 14.0 9.8 13.7 19.0 16.7 5.2 19.5 (m) 1.79 1.77 1.63 2.47 2.64 4.19 1.78 0.73 0.73 1.34 1.36 0.69 0.65 1.67  Observed W (m) 11.9 12.5 15.1 17.1 20.5 48.0 11.7 6.1 6.1 8.9 9.3 6.8 3.3 12.7 ‘  Modelled with Drained H/ Vbak / (m) Hcrgt Tent 2.36 100 0.04 3.08 1.00 0.02 2.63 0.95 0.02 2.60 1.00 0.02 3.89 1.00 0.02 4.83 1.00 0.02 2.38 0.65 0.04 1.18 0.50 0.03 0.98 0.46 0.05 1.82 0.38 0.03 2.19 0.33 0.09 1.26 0.53 0.02 0.93 0.28 0.01 2.23 0.61 0.02 1000 2 N/rn Saturated W H/ j (m) (m) 12.2 2.24 1.00 15.8 2.59 1.00 15.0 1.00 2.60 20.7 2.26 1.00 25.8 3.29 1.00 63.3 4.00 1.00 11.7 2.38 0.89 6.1 1.18 0.68 6.1 0.98 0.62 8.9 1.82 0.52 9.3 2.19 0.46 6.8 1.26 0.72 3.3 0.38 0.93 12.7 2.23 0.83 =  0.05 0.02 0.02 0.02 0.02 0.02 0.04 0.03 0.05 0.03 0.09 0.02 0.01 0.02  /rnjt  Vbank  29.7 13.9 32.4 22.3 50.0 9.4 7.0 11.1  -  10.8  -  (N/rn ) 2 32.8 10.0 11.9  rF  Estimated  Bank Vege tation Type T G T T G G T G T T T G G  Table 6.4. Results from the analysis of data from Chariton et aL (1978). The modelling with = 1000 N/m 2 forces the bank-shear constraint to be degenerate, and only the bank-height constraint is analysed. The estimated values of were obtained by varying trial values of Vcrjt until an agreement was obtained between W,,, and W . Reach 3 b 0 numbers 10 and 13 appear to be bank-height constrained and therefore no estimate of rrir could be obtained.  /s. This is to be expected as the smaller or lower discharge channels will develop 3 about 65 m values ofH that are lower than Hcrjt.  The remaining channels which are bank-height degenerate have the same optimal solution for drained and saturated bank conditions. It is assumed that the value of Tcrjg is unaffected by bank saturation. For these channels the modelled values are all narrower and deeper than their observed counterparts. This suggests that the values of  Thank  from the modelled channels are  greater than can be sustained by the observed channel. By reducing the value of rrgt, and therefore reducing the resistance of the banks to withstand applied shear, a wider and shallower channel will result. In this way the modelled channels can be brought into agreement with the observed geometries, and estimates of zrjt obtained. Also note that channels 5, 7, 8, and 12, which appear to be bank-height constrained when the value r  =  1000 N/rn 2 is used, are also much narrower and deeper than their observed  counterparts. For smaller values of  z,rit,  these channels will become bank-shear constrained and  . Therefore only channels 10 and 13 the value of Wmod can be brought into agreement with 0 W b  b and 0 are probably truly bank-height constrained as the values of W  obs T  lie within the range  defined by the limiting cases of fully saturated and fully drained bank conditions. 0 within a tolerance of ±1% for all channels with The value of TCrIt was varied until Wmod = W the exception of numbers 10 and 13. These estimated values of  are also shown in Table 6.4.  , with a mean value of 20.1 N/rn 2 . 2 The values range between 7.0 50.0 N/rn -  152  6.4.3 Effect of Bank Vegetation. Chariton et a!. (1978) have categorised the rivers in their data set on the  basis of bank  vegetation into channels with grassed (G) or treed (T) banks. The values of  obtained from  TCrjt  the previous analysis show a strong influence by the bank vegetation. The mean value of Thrit for 2 for the , and 29.8 N/rn 2 the channels described as having grassed bank vegetation is 10.4 N/rn treed banks.  The values of  Tcrjt  are plotted in Fig 6.8 and this indicates a strong division between the two  Vegetation Types. With the exception of one treed channel all of the treed channels plot above . This result suggests that the effect of 2 20 N/rn , and all of the grassed channels below 14 N/rn 2 the bank vegetation is to increase the value of  Thrit  either by binding the sediment by the root  masses, or conversely by affording protection of the bank and effectively reducing the value of Thank  acting on the bank sediment. Furthermore the bank vegetation may bind the bank sediment  as to increase the stability of the banks with respect to mass failure. In this way the roots act as internal reinforcement, and have the effect of increasing the effective values of c’ and qS’ above the 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 stabiising the 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 be  153  discussed as it highlights several important aspects of the above analysis. In this study stability curves analogous to Fig 5.3(b) were constructed for the Long Creek drainage basin in northern Mississippi using averaged values of c,  Yt,  and qS which were measured during previous field  surveys (Thorne er a!., 1981). Stability curves were developed for drained bank conditions using the measured averaged values of c,  ,  and  and for “worst case” soil conditions which  reflects saturated bank conditions. For values of 9 greater than  600  the Osman  -  Thorne slab  failure analysis (Osman and Thorne, 1988; Thorne and Osman, 1988) was used to determine the stability curves that correspond to F = 1.0, from which Hcrjt can then be determined for any value of  The Osman  -  Thorne slab failure analysis is similar to the Culmann analysis  presented in Fig 5.1 and Eqns (5.3) and (5.4), but is modified to include the effect of tension cracks. 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) are reproduced in Fig 6.9.  With the exception of one data point, the points all plot below the “worst case” stability curve which represents saturated bank conditions at failure and corresponds to the CASE Ill example in Chapter 5 (Fig 5.4). Three of the points lie on, or very close to the saturated curve. This distribution 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 failed during the previous winter period. Since these channels are all stable with respect to the “worst  154  case” saturated bank conditions, this may at first seem puzzling. However the bank-shear constraint has not been directly addressed in the March et a!. study.  A bank can only be  considered 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 can explain 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 are of the banks of a small creek flowing across a beach at low tide. The bank material is well sorted fine to medium sand which is normally non-cohesive, but has developed apparent cohesion 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 is approximately 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 additional  loading of 75 kg (the author’s body weight) was applied, and therefore H  <  Both bank  stability constraints are satisfied.  Figs 6. 10(b-d) are a sequence of photographs that were taken at a site approximately 20 metres downstream from the location in Fig 6.10(a). This downstream location was experiencing rapid lateral channel shifting, and the bank is unstable. The time between each photograph is of the order of 10 seconds. At this location the channel bank is approximately 100 mm high, much less  155  than at the stable upstream location.  In Fig 6.10(b) the shallow, supercritical flow is beginning to undercut the vertical bank. In Fig 6.10(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 be preserved at low flows, the bank-height constraint would appear satisfied as the bank is much lower than the stable, vertical bank upstream. However the bank at this location is clearly unstable as the bank-shear constraint is not satisfied, that is  Zbank >  z, and bank is being  undercut which leads to eventual mass failure. Therefore the primary erosion mechanism is in fact fluvial erosion of sediment from the toe of the bank, and the observed mass failure is only a secondary effect.  March er a!. do not explicitly recognise the requirement of the bank-shear constraint, although they do state that the observed H 8 combination may have been different at the time of failure -  and 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 be considered to be stable when both the bank-height and bank-shear constraints are satisfied.  156  6.5 BANKFULL MODEL: VARIABLE-CHANNEL-SLOPE The model will now be modified to allow for variable channel slope. In this formulation the channel slope S will be treated as a dependent variable. The sediment discharge capacity of the channel at the bankfull discharge Gbf will be used as an independent variable. The continuity and  bank 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 independent variable.  The objective function is modified to:  maxf(Jd,],O,S)=1l  (6.6a)  where:  Gb  (6.6b)  PQbf  since Gb 1  p,  and  Qbf are  all prescribed in this formulation, the maximization of i is equivalent to  157  a minimization of S.  The flowchart for the modified optimization model is shown in Fig 6.11, and the source code for the computer program in Appendix B. The only difference between Fig 6.11 and the flowchart for the previous formulation in Fig 6.1 is addition of the bedload constraint which must be satisfied before assessing the objective function. The bedload constraint is satisfied by varying S for trial values Of Pbed.  An example of the variation of i with Pbed is shown Fig 6.12. Each point on the solution curves in Fig 6.12 satisfies the continuity, bank stability, and bedload constraints. As with the fixed slope model the objective function curve is smooth with only a single global optimum. Also shown in Fig 6.12 is the variation of 5, which indicates the optimal solution corresponds to a minimum slope condition.  6.5.1 Noncohesive Bank Sediment The variable slope optimization model will now be run using the data from Andrews (1984) and Hey and Thorne (1986). The value of Gbf was calculated from the observed geometry using Eqns (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 vegetation  on the channel geometry, including the variation of S. As in Section 6.3.2 modelled values obtained assuming çS  =  40° are assumed to represent the unvegetated channel dimension. The  results are summarised in Table 6.5.  158  The results summarised in Table 6.5 indicate that as the density of the bank vegetation increases the observed channels become progressively narrower, deeper, and less steep. The observed and 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 bedload constraint is satisfied by reducing the channel slope.  Table 6.5. Ratios between the observed and modelled values of W, Y, and S for variable slope model with çb 400. Vegetation Type I II III IV Thin Thick  5 I Wmod b 0 W  Mm 0.59 0.43 0.23 0.20 0.70 0.16  Mean 0.98 0.80 0.70 0.55 1.03 0.54  5I b 0 Y  Max 1.84 1.35 1.17 0.97 1.51 0.78  Mm 0.64 0.81 0.88 1.01 0.75 1.16  Sobs / Sm€jj  Ymoci  Mean 1.05 1.19 1.39 1.67 1.00 1.37  Max 1.43 1.71 2.61 2.60 1.24 1.59  Mm 0.84 0.81 0.66 0.68 0.88 0.86  Mean 0.99 0.93 0.90 0.85 0.98 0.95  Max 1.26 1.08 1.03 0.96 1.10 1.38  This effect of the bank stability will now be demonstrated using Reach 13 from Hey and Thorne. 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 modelled and observed channel widths. The output together with the observed channel dimensions are shown in Table 6.6. Note the large influence that  exerts on the channel geometry, including  the channel slope.  The results of this analysis indicate that W is most sensitive to variations in the bank stability,  159  followed by  Y, with S the least sensitive. For the combined Vegetation Type IV and thick  vegetation channels the channel widths, depths and slopes are in the order of 0.5, 1.6, and 0.9 times their respective unvegetated channel dimension.  Table 6.6. Effect of SL on Reach 13 from Hey and Thorne (1986). The effect of increasing the value of is for the channel to become narrower, deeper, and less steep.  Channel Observed Modelled qS=40° Modelledq5,=73.l°  Surface Width (m) 18.4 91.2 18.4  Mean Depth (m) 1.14 0.44 1.18  Channel Slope 0.0133 0.0183 0.0133  By forcing an agreement between Wmôd and W b, the values of Ymoci and 0  Vobs  must also show  close agreement as once Wmod is fixed, the value of Ym,ci is constrained by continuity. Similarly the value of S,,, is determined largely by the bedload constraint, when the channel width is forced to agree with the observed value the modelled channel slope must also show close agreement with the observed value. In other words when one of the values of any of the dependent variables is fixed, there is only one combination of the remaining dependent variables that can satisfy the constraints and fulfil the objective function.  Recall from Section 6.3.3 that when the value 5 ,ç.  =  40° was used to model reach 13 with the  fixed-slope-model, the computed surface width was equal to 73.5 m, in contrast to 90.0 m with the variable-slope-model. The larger value of Wm obtained with the variable-slope-model is  160  due to the increase in the channel slope to S = 0,0183 from S  =  0.0133 in order to satisf’j the  bedload constraint. This illustrates the interrelationship between the dependent variables, and that the adjustment of one variable such as W, cannot be viewed in isolation from the adjustments of the other dependent variables.  6.5.2. Cohesive Bank Sediment The model was run using hypothetical data; firstly constant; then the value of  Qbf  Qbf  was varied, and the value of Gbf held  was held constant and Gbf varied. For both analyses the  50 following values for the independent variables were used: D 0,1 m, ksb  nk  =  0.1 m, c’  =  10 kN/m , 3  =  20 kNIm , ç’ 3  =  =  50 = 0.025 m, ksbed = 0.075 m, d  25°, and  rjt  =  25 N/rn . When Qbf 2  was varied the value Gbf = 5 kg/s was held constant, and when Gbf was varied the value Qbf = 100 rn /s was held constant. The results are presented in Figs 6.13 and 6.14. The key result 3 from this analysis is the change in the active bank stability constraint.  For small values of Qbf approximately less than 100 m /s the optimum geometry is bank-shear 3 constrained and bank-height degenerate, and those in excess of 250 rn /s or so the channel 3 becomes bank-height constrained and bank-shear degenerate (Fig 6.13(c)). Between about 100 /s the channel is actively constrained by both the bank-shear and the bank-height 3 to 250 m constraints.  When Qbf was held constant and Gbf varied, a similar result was obtained whereby the channels with values of Gbf less than about 3.0 kg/s are bank-height constrained and bank-shear  161  degenerate, while those in excess of 5.0 kg/s are bank-shear constrained and bank-height degenerate (Fig 6.14(c)). Within the range between 3.0 to 5.0 kg/s the modelled channels both bank-height and bank-shear constrained.  To explain this change in the active constraint, first consider the case where  Qbf is  held constant  so the results do not become confused by the different channel sizes which are associated with different  Qbf values.  For low values of Gbf the values of S, ; and  Thank  are at their minimum, and  Y is a maximum (Fig 6.14). The channels corresponding to the low values of Gbf are bankheight 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 increase in both the mean velocity, U and W. The depth decreases at a slower rate than the value of S increases, and therefore the values of r and continues until the value of  Thank  Thank  becomes equal to  both increase with increasing Gbf This Tcrjt,  at which point the channels become  bank-shear constrained. The value of S continues to increase more quickly than the depth decreases, 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 (Fig 6.13 (c)), which is a feature of natural rivers that is well known from field observations. For small values of Qbf the channels are bank-shear constrained due to the combination of the high and  Thank  T  values that are associated with steep channel slopes, in addition to the small values of  H which are well below Hcr:t.  162  As  Qbf  is increased the value of Y increases and S decreases, however the values of v and  rnk  remain initially constant because the channel is bank-shear constrained. Eventually the value of Y (and by definition H as H = Y for Qbf) increases to the point where the channel becomes bankheight constrained. For increasing Qbf beyond this point, Y tends to increase at a lower rate than the decrease in S and therefore the shear stress values decrease, and this ensures that the channel 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 equal /s. At this point H 3 to 2.95 m at Qbf= 100 m and approaches 12.7 m for  Qbf  Hrit. Yet for larger  Qbf,  the values of Y increase  s where H is still equal to Hcrjt. This is possible due 2,500 3 m /  to the change in 8 and the influence that this has on the value of Hcrjt. At value of 6 = 90° and Hcrjt and  =  Qbf =  s the 100 3 m /  s the value of 6 has decreased to 44°, m / 2.95 m. For Qbf = 2,500 3  has increased to 12.7 m. The influence of 9on the value ofHcrit is evident from Fig 5.3  and Eqn (5.2).  The general conclusion from the modelling in this section is that channels with large values of S tend to be bank-shear constrained. The large S may be associated with large sediment loads, or small values of Qbf, or a combination of both. Bank-height constrained channels tend to be associated with low S values.  163  6.6 SUMMARY  Bank stability has been shown to exert a strong control on the optimal geometry of alluvial gravel-bed rivers. The bank stability constraint was formulated for noncohesive channel banks and 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 model performance. The results are in good agreement with the results that Andrews and Hey and Thorne obtained from empirical regime analyses. The influence of the bank vegetation appears to stabilise the bank sediment, allowing the banks to withstand higher shear stresses. In vegetated channels this results in channels that are narrower, deeper and less steep than their unvegetated counterparts. The change in W is the largest, followed by Y, and the smallest is in S.  It was proposed that the effect of bank vegetation on channels with noncohesive banks can be represented by  ,  which was found to increase consistently with the bank vegetation density.  Model formulations were also developed for channels with cohesive banks. The model was tested on data from Charlton et al. (1978). From this analysis it was shown that channels can be either bank-shear or bank-height constrained. The values of  TCr,t  was shown to be higher in  channels with treed banks, than for there grassed counterparts.  Modelling results using hypothetical data indicate that bank-shear constrained channels are associated with large values of S which can result from large imposed sediment loads, and/or  164  small values of  Qbf  Conversely bank-height constrained channels tend to be associated with  lower values of S which can result from large values of Qbf, low sediment loads, or a combination of both.  In general only one bank stability constraint is active in channels with cohesive banks, and recognition of this is essential when assessing channel stability.  165  Figure 6.1. Flow chart for the bankfull:fixed-slope optimization model.  166  20  18 oJ  E  16  14  12  20  40  60  80  e(°) bank T  bed  Figure 6.2. Variation of Thank and Tbed with 6 Each point on the curves satisfies the continuity constraint for a constant value of Phd.  167  40  0.2 C.)  0.18  C.)  0.16  t C  Cl,  30  0.14 10  20  0.12 0.1  10  0.08 0.06 0.04  20  60  40 bed  80  0 100  (m)  1  Figure 6.3. Variation of i and 6 with Fbed. Each point on the solution curves satisfies the continuity and bank-stability constraints.  168  (a)  100 50 20 V a,  :i0 V  5  2. Observed Width (m)  (b) E a,  U V a,  a)  V 0  0.3 0.5 1 Observed Depth (m)  Type I o  Perfect Agreement I  Thin •  Figure 6.4. Comparison of Modelled and observed values of W and Y for rivers with noncohesive banks and low densities of bank vegetation. The data points denoted “thin” are from Andrews (1984), and those denoted “Type r’ are from Hey and Thorne (1986). The modelled values were calculated using the bankfull: fixed-slope optimization model.  169  (a)  100 ro  20 .1o 0  5  22 Observed Width (m)  (b)  0 .,  2-  .  0  0D*  E  • 4.0  a)  0 •  D E0.5 a) oO.3  0 ci 0  0  0 ci •  I  I I  0.2 I  —  0.101  ——I  0.2 0.3 0.5 1 Observed Depth (m)  2  3  I  Perfect Ripe I Type II Type III Type IV Thin Thick Agreement o  *  C  Figure 6.5. Comparison of Modelled and observed values of W and 7 for rivers with noncohesive 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 optimization model.  170  (a)  100  -50 4-  20 .  V 0  Observed Width (m)  3  (b)  2  E 4-  1  a)  0.5 a) o  V  0.3 0.2  0.101  1 0.2 0.3 0.5 Observed Depth (m)  2  Type I Type II Type III Type IV Thin Thick 0  3  Perfect Agreement  *  Figure 6.6. Comparison of modelled and observed values of W and 7 for rivers with noncohesive banks and for all categories of bank vegetation density using the mean values from 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 calculated using the bankfull: fixed-slope optimization model.  171  70  (a)  60 -c 0 .  40  30  20 10 0  (b)  0  60  20 30 40 50 Observed Width (m)  10  70  5  4  C Cu  3 2  0, 0) V 0  1  00  1  2  3  4  Observed Mean Depth (m) Bank-Shear Constrained Bank-Height Constrained Drained Banks 4  Bank-Height Constrained Saturated Banks  Perfect Agreement  Figure 6.7. Comparison of Modelled and observed values of Wand Y for rivers with cohesive banks from Chariton et a!. (1978). The modelled values were calculated using the bankfull: fixed-slope optimization model for values of rrjt = 1000 N/rn . 2  172  60  T  50  C.j  0 E 4  Z  T 30  20  ----TT -  Mean value for grassed banks  10  eT  T----G  0—  I  0  2  4  I  I  I  6  .  G  I  I  8  10  I  I  I  12  14  Reach Number  Figure 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.  173  LU  w  I BANK ANcLE (DEcEES)  I  —  C  STASLEDURINGI992  Figure 6.9. Bank stability curves from March et a!. (1993). The lower “worst case” curve corresponds to the Case III example from Fig 5.4, and the upper curve for average soil conditions corresponds to Case I from Fig 5.4.  174  I  : . .  (a)  :  Z4  ir  A.__  -‘•  ‘—.  ..  (b)  Figure 6.10 (a) and (b). Photographs of stream bank erosion from a small creek flowing across a beach at low tide. The lens cap in the photographs is for scale and is approximately 45 mm in diameter. See text for discussion. 175  a ‘a  r  -  -:  (c)  a.  (d) -  a.  -•‘  Figure 6.10 (c) and (d). Photographs of stream bank erosion from a small creek flowing across a beach at low tide. The lens cap in the photographs is for scale and is approximately 45 mm in diameter. See text for discussion.  176  N  Figure 6.11. Flow chart for the banlcfull: variable-slope optimization model.  177  0.004  0.055 0.05  0.0035  C)  C) C)  0.045 0.003  0 cj  H C)  0.04 0.0025 0.035 0.002 0.03  C)  clD  0.025  0  20  60  40 bed  80  100  0.0015  (m)  77  Figure 6.12. Variation of 77 and S with Pbed. Each point on the solution curves satisfies the continuity, bank stability, and bedload constraints.  178  (a)  14  100  12 80 100  CD  60  3:  8. 6  40  4 20  2  0  50  100  500  200  1,000  0  2,000  0.0035  (b)  45  0.003  a) o. o  40  0.0025 0.002  0.0015  30  3  0.001 25 0.0005 20  0  50  100  Duai Constrained  Bank-Shear Constrained  (c)  500  200  :  1,000  2,000  Bank-Height Constrained  25  1  -i  0.g  -.-----------  0.8 0.7 0.6  H/H  -  bank’ crlt  -153  tbank  0.5 0.4 50  100  200  20  500  1,000  2,000  Q (me/sec)  Figure 6.13. Variation of the optimal values of selected dependent variables as a function of rjt, Thank / Zcrit, and z. The value of Qbf is constant and 0 Qbf (a) Wand Y. (b) S and i (c) HI H /s. The active bank-stability constraint is indicated in Fig 6.13(c). 3 equal to 100 m  179  (a)  .  10 1  (b)  2  51020501  50  0.004 0.0035  Slope  -  25  0.0005  0.5  1  2  5  10  20  50  DuConsqed Ban  Bank-HeghtConstrairied  honsJned  G bf (kglsec)  Figure 6.14. Variation of the optimal values of selected dependent variables as a function of Thani I Tcrit, and Thank. The value of Qbf is constant and Gbf (a) W and Y. (b) S and z (c) H I /s. The active bank-stability constraint is indicated in Fig 6.13(c). 3 equal to 100 m  180  CHAPTER 7  FULL MODEL FORMULATION  7.1 INTRODUCTION  In this chapter the final formulation of the optimization model is presented. The equilibrium geometry 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 modified Parker (1990) surface-based bedload transport relation.  7.2 MODEL FORMULATION The full model formulation is similar to the bankfull:variable-slope model presented in Chapter 6 except that the sediment transporting capacity of the channel will be estimated using the Parker (1990) surface-based bedload transport relation, rather than the Einstein-Brown relation. The Parker surface-based relation was modified in Chapter 4 to apply to a natural channel with variable flows. This relation can be inverted to calculate the composition of the bed surface, or armour layer. The mathematical formulation of the full model is presented below.  181  7.2.1 Independent Variables In this analysis the independent variables whose values are presumed to be known are the fill range of flows and their durations (which can be represented by a flow-duration curve), the grain size mixture of the imposed bedload sediment, Gb, D5Obank,  $,  for noncohesive banks, and c 6’,  Yt,  and  kjb& ksbk,  the bank stability variables  for cohesive banks, and the constants  g, p, s, v, y  The flow-duration curve and the grain size distribution are subdivided into a finite number of intervals which are assigned representative values. The flow-duration curve is divided into m intervals and the grain size distribution curve into n intervals (See Figs. 7.1 and 7.2). Because both the flow-duration curves and the sediment size distributions are typically approximately log-normal (Shaw, 1988 p. 276; Parker, 1990) the representative value for each interval is given by the geometric mean:  where  Q’  Qj=JQ,1.Q  (7.1)  =JD’.D 1 D  (7.2)  and D are the geometric mean values of intervals i and  j  respectively, and the  superscripts 1 and u indicate respectively the lower and upper bounds of the interval. The values of i and j range from 1 to m and 1 to n respectively. For each interval the proportion of the total is determined. For the flow-duration curve the proportion of the total flows which fall  within a specified discharge interval is denoted by i. Similarly for the grain size distribution 182  curve, the proportion of the sediment by volume, or the fraction content within a specified grain size interval is denoted F. Summations of  i  and F over all i andj respectively are both equal  to 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 the significant duration is assumed to be one year, and therefore the Gb represents the mean annual bedload supply. The units of Gb will be kg/y, and the conversion factor, T will have a value equal 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 a dependent variable. This will be investigated in Section 7.6.  7.2.2 Dependent Variables The primary dependent variables which are to be solved are Pbed,  Pbank,  O S, and the grain size  distribution of the bed surface which will be represented by . 50 Other secondary dependent D variables which include  Rh, (J W, and  Y are readily obtained once the primary dependent  variables have been determined.  183  7.2.3 Objective Function The objective function in the full-model formulation is give by:  maxf(Jd,Pb,6,S,DSO)  where the coefficient of efficiency  i  (7.3a)  is given by:  77=  Gb m  Tpp,S  where  =  Gb =  (7.3b)  —  ‘°  the mean annual flow rate with units of 3 m / s. The total stream power which is  represented by pQS for steady, uniform flow is modified in the denominator of Eqn (7.3b) for the full range of flows as given by the flow-duration curve (Fig 7.1). The total stream power expended over one year per unit channel length is given by I’ p  Since the values of Gb and  S.  are imposed, a maximization of i is equal to a minimization of S.  7.2.4 Constraints The constraints for the full model formulation are given below.  7.2.4.1 Continuity The continuity constraint is unchanged from previous formulations and requires that the discharge capacity of the channel is equal to the value of the bankfull discharge, Qbf: 184  (3.24)  UA=Qbf  where A is the channel cross-sectional area at bankflill, and U is the mean channel velocity at bankfull which is given by: (3.35)  the value offis given by Eqn (3.19).  7.2.4.2 Bedload  The bedload constraint requires that the bedload transporting capacity of the channel be equal to the imposed sediment load Gb:  Fg=G  (4.54)  g=rspq  (4.55)  where:  where  qb  is given by Eqn (4.53). The value  qb  is the average volumetric bedload transport rate  /s. The units of Gb and 2 per metre channel width, and has units of m  gb  are kg/y and kg/y/m,  respectively.  The bedload constraint implicitly contains the “equal mobility constraint” which requires that all  grain sizes be transported equally over the one year duration (see Chapter 4).  185  7.2.4.3 Bank Stability The bank stability constraint can be formulated for both noncohesive and cohesive banks.  The noncohesive bank stability constraint is given by:  5 kfl1_s: y(s—i)D  (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-height constraint, and the bank shear constraint, which are, respectively:  1 HH  rbflk  crit 1  (5.7)  (5.8)  7.2.5 Optimization Scheme The optimization flowchart is shown in Figure 7.3 and is unchanged from Fig. 6.11. However the bedload constraint is substantially more complex than previous formulations and is shown in Figure 7.4.  186  In 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 be known. 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 F are calculated which satisfy “equal mobility” using Eqn (4.52) from which updated values of Dsg and o are obtained. This is repeated until convergence is attained on the values ofD and o. From the values of13. at convergence, the value ofD 50 can be readily determined.  7.3 VARIABLE FLOWS  The Parker surface-based bedload transport relation was modified in Chapter 4 to accommodate the variable flows which are inherent in natural river systems. The armour layer develops such that “equal mobility” applies to the total load transported over a significant duration (which will be taken as 1 year) as a result of the total range of flows. The size distribution of the annual transported sediment load is constrained to be equal to the subarmour sediment.  To illustrate the process of modelling a range of flows a hypothetical channel with noncohesive banks 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 m /s. The grain size distribution of the bedload and 3  subarmour sediment is shown in Fig. 7.2, and both have values of = 50 0.025 m and o = 0.43. d  The value ag is the geometric standard deviation which is given by:  187  ir  2 (Dl  usg=a[logLJj F  The values of  and  ksbk  (7.4)  will both be set equal to 0.40 m, D5Oba,, set equal to 0.07 m which  is about the anticipated value of , 50 and ç = D  400.  The total annual imposed sediment load is  2.5 X 106 kg/year. The value of the bankfiill discharge,  Qb.,  is set equal to 85 3 m / s which is  equalled or exceeded 1.7% of the time. This value of Qbf is not arbitrary but it will be shown in Section 7.6 that it corresponds to an optimum. The flows that exceed  Qbf  are set equal to  Qbf  following the assumption of an infinitely wide flood plain. This assumption is discussed in Section 3.3.  The optimum values of W,  7, S,  and D 50 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 these  modelled values are typical for natural gravel rivers with little bank vegetation (eg. see regime equations and field data from Hey and Thorne, 1986).  The sediment transport rate Gb ’ for each value of Q’ is shown in Fig. 7.5(a). Here the prime (‘) 1 is used to denote the rate in kg/s, and Gb, without the prime denotes the total transported in one year, 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), while  188  the 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 is shown in Fig. 7.6 and illustrates the definition of “equal mobility” as applied herein. The value of dSOJoad varies with discharge; the lower discharges are associated with values of d5Oload less 50 The net result is that d than , 50 while higher discharges result in values of d5Oload greater than . d 0 Note that . 5 over the entire year the value of dSOIoad for the total annual bedload is equal to d s the median grain diameter is constant. For these low m / below a discharge of about 35 3 discharges the bed surface is essentially immobile (Fig. 7.5(a)), and in most natural rivers any sediment transport would be restricted to “sandy throughput load”. In Oak Creek the sediment s had a mean grain diameter m / trapped for discharges less than the threshold value of about 1 3 of about 2 mm (see Fig. 4.12). Once this threshold value was exceeded the value of d5Oload showed a consistent increase with discharge.  7.4 ADJUSTMENT OF THE BED SURFACE COMPOSITION In previous model formulations (Chapter 6; Millar and Quick, 1993; Miflar, 1991; as well as  White et a!., 1982; Chang, 1979, 1980) the armour layer grain size distribution as represented by D 0 was considered to be a fixed independent variable. In the full-model formulation the  grain size distribution of the armour layer is able to adjust by using the Parker (1990) surface50 will therefore now be treated as a dependent variable. based transport relation. The value ofD  189  The degree of armour layer development will be represented by the value of 650 which is the ratio 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 layer development, to greater than 6. Most rivers have values of 650 around 3 when measured at low flows. It is assumed in this thesis that the composition of the armour layer which is observed at low flows is maintained at high discharges (Sect 4.4.1). In addition to , 50 a characterisation of D the armour layer should also include a measure of the dispersion of grain sizes about the median value, however this is of secondary importance and will not be considered herein.  7.4.1 Sso  -  50 T*D  Solution Curves  Parker (1990) has shown that for an imposed bedload grain size distribution a solution curve can be calculated to predict the variation of D with  sgo. 5 q  In this section 5 o  -  50 T.D  solution  curves, which are analogous to Parker’s, will be developed to demonstrate the adjustment of  the armour layer. The 7.7. For values of  850  -  50 TD  solution curve for the Oak Creek sediment is shown in Fig.  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’D 50 greater than 0.03 a mobile armour is developed (Parker, 1990). The value of 8o is seen to decrease as 50 T’D  50 in excess of 0.3. increases, and eventually disappears for very large values of V*D  190  The value of ö observed at Oak Creek is about 2.2 which corresponds to a value of equal to about 0.06. The bed surface at Oak Creek, as with wide range of shear stresses. Therefore the value  50 r*D  =  any  50 v”D  natural river, is subject to a  0.06 for Oak Creek can be considered  to be the “dominant” or “channel forming” dimensionless shear stress which is analogous to the dominant discharge concept. This dominant dimensionless shear stress is likely to correspond to bankfull, 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 layer does not change appreciably despite the daily and seasonal variation in the flows and shear stresses.  The grain size distribution curves for the Oak Creek subarmour and armour sediment are shown in Fig 7.8. These values are from Parker (1990) and exclude the sediment less than 2.0 mm in diameter. The curve for the subarmour sediment has been slightly modified at the upper end because Parker reports that 0% of the subarmour sediment sampled was in the range 102203 mm, whereas 8% of the armour layer falls within this range. The upper end of the subarmour curve was smoothed to approximate a log-normal distribution with the maximum grain  size equal to 203 mm. This modified curve now has 3% of the subarmour sediment in the  range 102-203 mm.  2 was calculated The composition of the armour layer for an imposed bed shear stress of 45 N/rn using Eqn (4.44). The selected value of the bed shear stress gives a value  =  0.06 and  50 match. The results are plotted along with the therefore observed and calculated values of D observed grain size distribution in Fig. 7.8 and there is an excellent match. However the Parker  191  surface-based bedload relation that is used to calculate the composition of the armour layer was developed from bedload transport data collected on Oak Creek, and therefore the good match is not surprising.  As an independent test of the Parker theory the experimental data of Dietrich et aL (1989) will be used. Dietrich et at. performed three flume experiments to investigate the effect of sediment supply on the development of the armour layer. They initially started with very high sediment feed rates which at equilibrium resulted in no armour layer development (8o = 1). The sediment supply rate was progressively reduced and the bed surface was observed to coarsen to values of 850  =  1.16, and 1.32. The  850  5 V*D  solution curve was calculated using the sediment feed  grain size distribution, and is shown together with the three observed armour-layer values in Fig. 7.9. There is close agreement with the theory for two of the points, however the data point corresponding 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 range of data. In particular the reduced hiding function (Eqn. 4.36) was developed for r’j 50  0.05.  At this value the armour layer is only moderately mobile. However for higher sustained shear stresses where the bed surface is much more mobile and the value of öo approaches 1, it is unlikely that Eqn (4.36) would still be valid. The results from the analysis of the data from Dietrich et a!., which is admittedly based only on three data points, suggest that the Parker , say up to a maximum of 0.07. Beyond 50 theory is applicable only up to moderate values of z*D  this value the mobility of the bed sediment and the hiding relations appear to be considerably different from those observed at Oak Creek. Fortunately natural gravel rivers generally have  192  bankfull values of  50 T*D  less than 0.07, with most falling in the 0.03  -  0.06 range where the  Parker theory can be applied.  7.4.2 Effect of Sediment Gradation on Theoretical ö  -  50 r’D  solution curves were calculated for a range of sediment mixtures from  uniform to poorly sorted which are shown in Fig. 7.10. The sediments all have the same median grain size (d 50  =  25 mm). The geometric standard deviation, 0 g, was used as a measure of the  dispersion about . 50 The resulting d  650  -  50 V*D  curves are shown in Fig. 7.11. The sediment  gradation 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 of 50 is approximately equal to d D 85 in this example.  Table 7.1. Summary of sediment size parameters from Figs 7.10 and 7.11. The values of Ug and d 90 are from the subarmour sediment, and 850 and D 50 are from the computed armour layer. The value ofD 50 is approximately equal to . 85 d (Jcg  0 0.28 0.43 0.50  d 8 5 (mm) 25 45 75 100  1 1.6 3.1 4.4  D 5 0 (mm) 25 40 78 110  7.5 EFFECT OF SEDIMENT LOAD The effect of increasing sediment load on the channel will be modelled in this section and the theoretical results compared to experimental and field observations, and qualitative relations obtained from these observations.  193  The effect of a 10-fold increase in the imposed sediment load from 2.5 X 106 to 2.5 X i0 7 kg/year on the optimal hydraulic geometry for the set of independent variables described in Section 7.3 is shown in Fig. 7.12(a) and (b). The values of W and S increase in value with increasing sediment load by 32% and 56% respectively, while the values of 7, and ]3 show a decrease of 24% and 15%.  These modelled adjustments above are in general agreement with observed adjustments. Recall the 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 . 50 D  Since the value of Qd is assumed to be constant (cf Section 7.6.1), Formula (1.1) indicates that an increase in Gb will produce an increase in the value of S, and a decrease in D. The decrease in the value of Tho with increasing sediment load was discussed in Section 7.4 and is supported by numerous experiments and field observations (eg Dietrich et at., 1989; Lisle and Madej, 1989; Kuhnle, 1989).  A similar qualitative relation proposed by Schumm (1969):  194  W2S  where 2= the meander wavelength, and  =  cxG,,  (1.3)  sinuosity, which is equal to the valley slope divided  by the channel slope, S / S.  Formula (1.3) was derived from field observations, and indicates that an increase in Gb will result in an increase in W, S, and 2, and a decrease in Y and equivalent to a decrease in  .  By definition an increase in S is  as the value of S is assumed to remain constant. The adjustment  of 2 is not considered in the optimization modelling.  For an increase in the imposed sediment load the results obtained from the optimization modelling show good qualitative agreement with the proportionalities developed by Lane (195 5a) and Schumm (1969). These relations are widely used as guidelines for interpreting and predicting river adjustments, and the agreement between the modeffing results, and the above qualitative formulas lend general support to the optimization approach.  7.5.1 Valley Slope Constraint The adjustments to increasing sediment loading modelled above do not consider the valley slope constraint which represents a physical bound that defines the maximum channel slope that can be attained over engineering time scales. The valley slope is considered to have developed over geologic time, and therefore over engineering time scales can be considered to be an independent variable.  195  The initial hypothetical channel from Section 7.5 with a slope of 0.00426 and a sediment transporting capacity equal to 2.5 X 106 kg/year, would require a value of adjust to an imposed load of 2.5 X  1.57 if it were to  kg/year which requires an increase in the channel slope  of 57% to 0.00668 (Fig. 7.12(b)). The 62 rivers listed in Hey and Thome (1986) have an average sinuosity of only 1.34, and therefore the majority of these could not increase their channel 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 therefore determines the maximum sediment load which can be accommodated by a river system and still maintain a stable equilibrium channel. For example if the valley slope for the river system being modelled in Section 7.5 were 0.0056, then from Fig. 7.12(b) it can be seen that the maximum imposed load that can be maintained is about 1.0 X i0 kg/year. For a sediment load in excess of this value the channel cannot develop the required slope and therefore sediment continuity cannot be maintained. An aggrading, unstable, and possibly braided channel would be expected.  7.6 BANKFULL DISCHARGE AS A DEPENDENT VARIABLE Up to this point in this thesis, and generally throughout the literature, the value of Qbf is considered to be an independent variable which is a function of the catchment hydrology. This was questioned in Section 3.3. The optimization model will now be used to show that there is an optimum value of Qbf which suggests that, in the case of a river with an active floodplain, the  196  bankilill discharge should more correctly be considered a dependent variable. This analysis does not apply to incised channels.  The equilibrium geometry for the hypothetical channel described in Section 7.3 was calculated for a range of Qbf values from 55 to 155 m /s. The equilibrium values of S for the various 3  Qbf  values are shown in Fig 7.13(a). It is evident that an optimal value of Qbf which corresponds to  a minimum value of S (and hence maximum  ii),  occurs at about  Qbf=  85 m /s. 3  The reason for the existence of this optimum value of Qbf can be realised from examination of the transport rates and total annual loads transported over each flow interval. Contrast the transport rates for values of Qbf equal to 65 and 135 m /s which are given in Table 7.2. For 3 flow interval  /s the sediment transport rates are 0.84 and 0.26 kg/s for 3 Q which equals 59.8 m  each channel respectively. Over one year value of Qbf  =  Q transports  540,000 kg or 21.6% of the total for a  65 m /s, while for Qbf = 135 m 3 /s the total transported over one year by Q4 is 3  only 170, 000 kg, or 6.8% of the total. The same flow in the smaller channel has a higher rate of transport because the depth of flow and therefore the value of Thed are greater. For Qbf /s the depth of flow and value of 3 m  Thed  for flow  Q  =  65  are respectively 1.18 m and 46.1 N/rn , 2  while for a value of Qbf = 135 m /s the respective values are 0.85 m and 35.6 N/rn 3 . 2  As the value of Qbf increases the sediment transport rate at bankfull also increases. This is a result of the greater discharge as well as the wider channel bed across which bedload transport can occur. Compare the bankfl.ill sediment transport rates in both channels. For Qbf = 65 rn /s 3 the bankflill transport rate is 1.13 kg/s, while for Qbf  197  135 m /s it is 6.26 kg/s. However as Qbf 3  00  5  4  I 2 3  1  ms/s 8.1 30.4 48.1 59.8 65.0  0.753 0.141 0.050 0.020 0.037  Pi  Qbf  Qbf=  =  65 m /s 3 Y Gb’ kg/s m 0.04 0 0.81 0.026 1.04 0.395 1.18 0.839 1.23 1.13  Table 7.1 Comparison of flow parameters for  Gb kg/y 17.0 1.2 X 6.2 X 5.4X i0 5 1.3 X 106  65 3 m / s and Qbf  I 2 3 4 5 6 7 8 9 10 11 12  I m / 3 s 8.1 30.4 48.1 59.8 69.8 79.8 89.9 99.9 109.9 119.9 129.9 135  135 3 m / s.  0.753 0.141 0.050 0.020 0.012 0.008 0.004 0.003 0.003 0.002 0.001 0.004  Pi  =  135 m /s 3 Y m 0.27 0.59 0.76 0.85 0.93 1.00 1.07 1.13 1.19 1.25 1.31 1.34  Gb’  kg/s 0 0.00 1 0.047 0.264 0.557 0.880 1.33 1.98 2.83 3.96 5.39 6.26  Gb  kg/y 0.60 4900 74,000 1.7X 2.1 X 2.1 X 5 1.8X10 2.1 X 2.2X 2.OX 5 2.2X10 8.1 X  increases the duration which the bankflill discharge is equalled or exceeded also decreases. s the duration of the bankfull flow decreases from 0.0365 to 0.0041. m / From Qbf= 65 to 135 3  Therefore the optimal value of Qbf results from the opposing tendencies of high sediment transport rates for low flows in the smaller channels, and higher effective discharges and greater bed widths which combine to produce larger transport rates in the larger channels.  7.6.1 Effect of Sediment Load on Bankfull Discharge The effect of sediment load on the channel geometry will be used to demonstrate evidence that Qbf  should be considered a dependent variable. It was shown in Fig 7.13(a) that for an imposed  bedload of Gb  =  /s. The 3 2.5 X 106 kg/year the model returns an optimal value of Qbf = 85 m  7 kg/year are shown in Fig. 7.13(b) and indicate an results for an imposed load of 2.5 X i0 s. Therefore the value of Qbf is seen to adjust together with m / optimal value of about Qi,= 135 3 the 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 agreement  with the model predictions. Williams (1978) suggests that the recurrence interval (and hence the magnitude) of the bankfull discharge is possibly affected by the sediment load. It is an interesting 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 the effect on the channel width of a 10 fold increase in sediment load from Gb  199  =  2.5 X 106 kg/year  using the independent variables from Section 7.3. The variation in Wassuming a constant value of Qbf  =  85 m /s, as well as variable 3  Qbf  are shown. A much wider channel is predicted  assuming a variable Qbr  It can be seen from Figs 7.13(a) and (b) that the optimum value of S, and hence  i,  is relatively  insensitive to changes in Qbf For instance it can be seen from Fig. 7.13(a) that a 55% increase in  Qbf from  55 to 85 m /s produces a decrease in S (or an increase in 3  i)  of only 3.4%. A wide  range of values of Qbf are therefore “near optimal”. It would seem reasonable that the “driving force” 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 channel to attain the optimum value would be correspondingly small. Therefore while a theoretical  optimal value of Qbf can be determined, in reality there is a wide range of values of Qbf which come close to satisfying the objective function, and therefore in natural systems it could be expected that the observed bankfull flows would be randomly scattered over a wide range about the theoretical optimum.  7.7 APPLICATION OF THE MODEL The formulation of the model allows it to be applied to any situation where there is alteration in the volume and size distribution of the imposed sediment load, the volume and timing of the flows, or the properties of the bank sediment. For example the construction of a dam will affect the sediment supply and flows. The sediment supply will be reduced dramatically, often effectively to zero directly downstream from the dam. The area of interest may be some distance downstream and the reduction of the sediment load from the dam may represent only a  200  fraction 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 total runoff 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 by logging or for agricultural development may increase the sediment production from the catchment, and yet may or may not affect the runoff to any great extent. Often the riparian vegetation is affected by these developments, and as has been demonstrated in Chapter 6, this can have a profound influence on the bank stability.  Regardless of the type of development or catchment disturbance, the input required for the model is a flow-duration curve, an estimate of the volume of sediment load and grain size distribution, and estimates of the bank stability parameters. Hydrologic modelling and sediment budget studies may be necessary together with field observations to determine the appropriate values to use as input to the model. These values may be difficult to measure in an intact system let alone to predict the values for a disturbed catchment. Estimates of the current sediment load of a stable river system can be obtained using the observed hydraulic geometry, which includes the grain size distribution of the bed surface, together with the measured or estimated flow duration curve.  The model can be used to perform sensitivity analyses to determine the effect of a potential development on the river channel where the post-development inputs cannot be accurately determined. For instance some studies have indicated that typical logging practises in coastal  201  B.C. and the Pacific north-west can result in an 8 to 10 fold increase in the yield of coarse bedmaterial sediment as a result of increased mass-wasting processes (eg Madej, 1978). The model can be calibrated using the observed hydraulic geometry, and then the sensitivity of the channel to increased sediment load can be determined.  When using this optimization model it must be realised that the optimum value is only a theoretical value which the river may show a tendency to adjust towards. The optimization model is only an adjunct to other techniques such as air photo and field monitoring of observed channel adjustments of the river of interest and nearby channels that may have been subjected to similar disturbances. Any modelling results must be tempered with sound engineering judgement, and must recognise the location of geologic controls that may limit the computed adjustment.  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 has not been addressed. The value of the modelling approach has relied on its empirical success in  predicting channel geometry. It seems that the general reluctance of the engineering community to more fully accept this approach to river adjustments is largely due to the “lack of a physical basis” for this approach.  Yang and Song (1979, 1986) have attempted to explain their minimum energy dissipation rate  and minimum unit stream power hypotheses by arguing from fundamental fluid mechanics precepts. Yang and Song suggest that many aspects of the behaviour of fluids, and not just  202  river channel adjustments, can be explained by their minimum energy dissipation rate hypothesis. However their explanations have not met with widespread acceptance.  The argument to be presented in this section will attempt to demonstrate that river channels display the tendency to move towards the optimal configuration as a result of random perturbations which exist in abundance in natural river systems. It is the asymmetry in river channel response to the random perturbations (which will be explained below) that drives the system towards the optimum.  Figure 7.15 shows a single solution curve that satisfies the continuity, bedload, and bank stability constraints for a particular set of independent variables. The optimal geometry is located at Point A. Any channels that are located in the area above the solution curve are over capacity with respect to transporting sediment, and have the capacity to transport sediment in excess of the imposed load. Similarly channels located in the area below the curve do not have the capacity to transport the imposed sediment load. The further a point is above or below the solution curve, the greater the sediment transport capacity of the channel diverges from the imposed load.  Consider a channel that is initially located at Point B. This channel is narrower and steeper than the optimum. Ideally this channel could remain indefinitely at Point B in apparent stability as it satisfies all of the constraints. However in nature there are abundant opportunities for the channel geometry, at least locally, to be perturbed from B. Consider a local widening which could result from an event such as tree falling into the river which acts as a locus for deposition,  203  from which a gravel bar is developed. This could deflect the current locally against the bank resulting in local bank erosion and widening of the channel to Point C. The channel is now, at least locally, over capacity, and will attempt to fulfil its sediment transporting capacity through bed and bank erosion.  It is a primary assumption throughout this thesis that bed erosion and degradation is negligible in comparison to bank erosion, and that bank erosion is concentrated along the outer banks of meanders. When the sediment transporting capacity of the channel becomes greater than is required to transport the volume of sediment imposed from upstream, net erosion will occur along the reach as the channel seeks to fulfil its sediment transporting capacity. Since the erosion is presumed to be concentrated along the outer banks of the meanders, this erosion will result in further development of the meanders, therefore causing a reduction in the channel slope. There may also be some limited channel widening. The net result is that the channel eventually returns to the solution curve at Point D. The result of the initial perturbation is therefore to drive the channel geometry closer to the optimum.  Perturbations which displace the channel geometry below the solution curve will produce a local channel geometry that has insufficient capacity to transport all of the imposed sediment load. 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 to envisage how significant changes would result.  204  It is therefore possible that an asymmetry exists whereby perturbations which displace the channel geometry above the solution curve result in a modification of the channel geometry which moves it towards the optimum. Perturbations which displace the channel geometry below the solution curve into the zone of under capacity will be damped out and have lesser effect on the geometry.  Once the channel reaches the optimum, any changes in the channel width will reduce the sediment transport capacity and these perturbations will be damped out. Any steepening of the channel 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 a reduction in the channel slope returning the channel to the optimal configuration.  The above explanation of the optimization process is admittedly incomplete. The role of the secondary currents is not considered. This explanation does indicate however, that the tendency for alluvial rivers to attain an optimum configuration is not a systematic process, but results from random heterogeneities in the channel.  7.9 COMPARISON WiTH OTHER NUMERICAL MODELS The modelling strategy developed in this thesis is fundamentally different from most other numerical models which have been developed to predict river adjustments. The most widely used 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) is the most well known. These models assume that the channel width remains constant and  205  changes 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 or degradation. Osman and Thorne (1988) and Thorne and Osman (1988) modified this approach to include a bank erosion algorithm for cohesive channels. Chang (1982) proposes that crosssectional changes, which include a reduction or increase in the channel width, can be determined for each time step by using the minimum stream power hypothesis. These models can be run until steady state conditions have been attained.  In contrast the optimization model proposed herein does not consider the transient, dynamic channel adjustments and gives only the final steady-state channel geometry. Changes in channel width are modelled, and the slope adjustments over engineering time scales are assumed to occur 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. Any differences that would arise between the models due to the different flow resistance and bedload transport equations used in each model are ignored. Two solution curves similar to Fig 7.15 are shown. The two curves in this case can be taken to correspond to two values of imposed bedload, the upper curve representing the higher value of Gb. Consider a stable channel located at Point A on the upper curve which is subject to a reduction in sediment load to the value represented by the lower curve. The optimization model predicts that the new stable equilibrium geometry will be given by Point B on the lower curve which represents a reduction in the channel slope and a decrease in the width. There is no indication of the pathway or the time required for the channel to adjust from Point A to B.  206  A fixed width model such as HEC-6 will return a progressive decrease in the channel slope due to bed degradation as the sediment transporting capacity of the channel will initially be greater than the sediment supply following the reduction. Assuming that the degradation is not terminated by the development of a static armour layer (which is almost certain to occur in a gravel river), the HEC-6 type models will eventually arrive at a steady state condition close to Point C on the lower curve. Since the degradation has not involved any changes in channel width the final channel must be deeper than the original and it is probable that the bank stability constraint will not be satisfied at steady state. However with the exception of the Osman Thorne model to be discussed below, these dynamic models do not consider the stability of the banks.  The Osman-Thorne model recognises that the bank stability and erosion must be addressed. In their model they calculate the net bank erosion in each time step and add this volume of eroded bank sediment into the sediment continuity calculation for each time step. The eroded bank sediment 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 steady state solution at some location between Points C and D on the lower curve. The exact location will depend upon the relative erodibility of the bed and bank sediment, for highly resistant banks the final solution will be close to Point C (slope adjustment only), and for easily erodible banks relative to the bed the final solution will be closer to Point D (width adjustment only).  207  The numerical model of Chang (1982) uses sediment routing to compute the change in the cross-sectional area of the channel, AA, which can be positive or negative depending upon whether the channel is aggrading or degrading. The minimum stream power (Section 2.2.3) is used 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 the bed. One feature of the model is that uniform flow is not assumed, and therefore the slope of the bed and the energy gradient are not necessarily the same. The dynamic optimization model of Chang predicts that the channel in Fig 7.15 will move along a direct line from Point A to Point B.  Despite the capacity for width adjustment, the numerical model of Chang does not contain algorithms for calculating the distribution of the boundary shear stress, or for bank erosion or deposition. For example the model formulation does not recognise that the rate of bank erosion is dependent upon bank shear stress and the erodibility of the bank sediment. Furthermore it is assumed that the channel adjustment is a continuous process of dynamic equilibrium, and that the channel maintains equilibrium at each time step. However in the static optimization model proposed in this thesis, while the channel is assumed to ultimately reach an optimum, the transient adjustments are not presumed to adhere to any principle of equilibrium.  Clearly the models appear to give very different results, particularly the optimization model and the Osman-Thorne model which both consider the bank stability and therefore one would assume 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 channel  208  narrowing in this case is not possible. The optimization model developed in this thesis does not explicitly include bank deposition in its formulation, however bank deposition is implied when the value of  falls below the critical value for bank stability since the banks of an optimal  channel develop at the limiting stability.  One other important difference between the optimization model and the dynamic models is the latter’s dependence upon the initial conditions. It is evident from Fig. 7.16 that the final solution for the dynamic models is dependent upon the initial conditions, different initial conditions will yield different final solutions. However the optimization model is independent of the initial conditions. It was argued in Section 7.8 that random perturbations in the course of a river channel’s adjustment were necessary for it to attain the optimal geometry. These random events are not accounted for in the dynamic models and are important as they have the effect of resetting to some degree the initial conditions. While final steady-state solution from the dynamic optimization model of Chang (1982) is not dependent on the initial conditions, the path and the time taken during adjustment are.  An interesting possibility would be to include a random component into the non optimization  dynamic models to see if there is a tendency for the channel to adjust to the optimal configuration.  209  7.10 AN EXAMPLE OF RiVER BEI{AVIOIJR WITHIN AN OPTIMIZATION FRAMEWORK  The optimization framework can be used to interpret the response of an alluvial river following river training activities. A solution curve calculated using the optimization model is shown schematically in Fig 7.17. The valley slope constraint is also indicated. The feasible part of the solution curve, that is the section of the curve that satisfies all of the constraints including the valley 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 commonly straightened. If the straightened channel is oriented parallel to the valley axis, it will have a slope equal to S. The channellized reach can be designed such that it possesses approximately the same sediment and discharge capacity as the natural meandering channel, and also has stable banks. A straight channel which is oriented parallel to the valley axis and also is designed so that is satisfies the discharge, bedload, bank stability and valley slope constraints is indicated at Point B. Note that this represents the narrower of two possible options; a wider channel that also 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 the reach the channel slope is increased, and therefore the channel is no longer at the minimum slope that corresponds to the optimal geometry. These straightened channelled reaches must usually be maintained as any slight channel curvature or irregularity is often amplified by the flow, and this can result in the development of meanders which cause a reduction in the channel  210  slope. The tendency of a straightened channellized reach to develop meanders is interpreted as the channel attempting to reestablish the optimum geometry.  7.11 SUMMARY  The complete model formulation was presented in this chapter. The model was formulated to use the fill range of flows, which are represented by a flow-duration curve, as inputs to the model. Furthermore the composition of the bed surface is able to adjust to the optimal configuration which is based on the maximization of the coefficient of sediment transport efficiency,  i.  This represents a significant advance over previous optimization formulations  such 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 of gravel rivers were demonstrated. The effect of sediment load on the equilibrium channel was shown to agree well with field observation. The model predicts that an increase in the sediment load will result in the channel becoming wider, shallower, steeper, with a decrease in the value ofD 50 of the bed surface.  The common assumption of the bankfull discharge as a fixed independent variable was examined. It was demonstrated that there is a value of Qbf for rivers with active floodplains that corresponds to an optimum. However the value of  i  is relatively insensitive to different Qbf  values, and it is probable that natural rivers show a range of randomly about this optimal value.  211  Qbf  values that are scattered  It is argued that natural channels tend towards an optimal geometry as a result of random perturbations which exist in abundance in natural rivers. These random events are not considered in the dynamic, numerical models of river channel adjustment. Furthermore the dynamic models all attribute changes in the channel slope to either aggradation or degradation, and do not consider changes in the channel sinuosity.  212  200  150  1100 50  0  0.001  0.01  0.1  1  Fraction of Time Discharge is Equalled or Exceeded  Figure 7.1. Flow-duration curve showing numerical approximation. The flow-duration curve in this example is subdivided into 18 discharge intervals. A maximum discharge value must be selected that is equalled or exceeded all of the time which is the maximum discharge on record, or an approximation.  213  100  80  60 a)  C LL  40  20  0  2  5  10  20  50  100  200  Grain Size (mm)  Figure 7.2. Sediment gradation curve showing numerical approximation. The sediment gradation curve is subdivided in this example into 7 sediment intervals.  214  N  Figure 7.3. Flow-chart for fhll optimization model formulation.  215  Initialize Trial D and a Values sg  Calculate Transport Rates For All Grainsizes  Calculate all F That  Use Calculated  Satisify Equal Mobility Eqn(4.52)  D and a Values sg  •  Calculate New D and a Values sg  Have and a, Values  sg Converged?  Calculate 50 Gb D  End Bedload Sub Function  Figure 7.4. Flow-chart for bedload subfunction. An iterative scheme is required to calculate the values of D and u.  216  (a) 2 a, t 0 Cl,  I 4-  a,  a,  Co  0  0  20  40 60 Discharge (m Is) 3  80  100  (b) 1 E + 07  1E+06 1E+05 -  1E+04  1E+03 1E+02 I-  1E+01 1 E + 00 Discharge (m /s) 3  Figure 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.  217  30  25 E N  U) a) .  20  a)  C  V  a,  C a) V  a,  15  10  0  20  40  60  Discharge  Figure 7.6. Variation  Of d5Oload with  80  100  Is) 3 (m  discharge. The value of the subarmour d 50 is 25 mm.  218  2.6 2.4 2.2  2 0  ‘01.8  1.6 1.4 1.2 1  0.01  0.02  0.03  0.05  0.1  0.2  0.3  T*  50 D  Figure 7.7. The calculated 850 solution curve for Oak Creek, Oregon. Data from Milhous (1972). The solution curve was calculated using the Parker (1990) surface-based sediment transport relation. -  219  100  80  I  60  C U  40  20  0  2  5  10  20  50  100  200  Grain Size (mm)  Figure 7.8. Armour and subarmour grain size distributions for Oak Creek, Oregon. Data from Milhous (1972). The calculated values were determined using the Parker (1990) surface-based sediment transport relation.  220  1.5  1.4  1.3 0 U,  ‘0 1.2  1.1  1 0.01  0.02  0.03  0.1  0.05  0.2  0.3  T*  50 D  Figure 7.9. Calculated öo TD 50 solution curve for data from Dietrich et a!. (1989). The solution curve was calculated using the Parker (1990) surface-based sediment transport relation. -  221  100  80  60 a)  C U  40  20  02  5  10  20  50  Grain Size (mm)  Figure 7.10. Sediment gradation curves for a range of crg values.  222  100  200  5  4  0  2  1  0.01  0.02  0.03  0.05  0.1  0.2  T*  D 5 0  Figure 7.11. Calculated c 50 curve for a range of ag values. The curves were calculated 5o TD 5 using the Parker (1990) surface-based sediment transport relation. -  223  60 Width Depth  55  E V  1.2  50 45 40  0.9  35  0.8 30 3.OE+06  5.OE+06  1.OE+07  2.OE+07  Sediment Load (kgly)  0.007  0.08  0.0065  0.075  0.006  0.07  0)  0.0055  0.065  0.005  0.06  0.0045  0.055  0.004  3.OE+06  5.OE+06  1.OE+07  2.OE+07  ,,  0.05  Sediment Load (kg/y)  Figure 7.12. Effect of sediment load on channel geometry. (a) W, Y. (b) S, . 50 The values D were calculated using the optimization model.  224  0.0048 0.0046  -  a)  0.0•  0.0042  —  0.004  60  140  120  100  80  Bankfull Discharge (m3/s) 0.0072 0.007  -  :: 0.0064  60  80  100  120  140  160  Bankfull Discharge 3 (m / s)  Figure 7.13. Calculated variation of S with Qbf (a) Gb = 2.5 X 106 kg/y. (b) Gb kg/y. The values were calculated using the optimization model.  225  =  2.5 X  70  Variabre  -.  bf  Constant  bf  60 2 .  •1  30 3.OE+06  I  I  5.OE+06  1.OE+07  2.OE+07  Sediment Load (kgly)  Figure 7.14. The calculated variation of W with Gb for constant and variable Qbf The values were calculated using the optimization model.  226  Over Capacity  0 C’)  C Solution Curve  Under Capacity  Width  Figure 7.15. Solution curve used to demonstrate channel optimization. The solution curve is shown schematically. Point A is at the optimum, Points B and D satisfy the constraints, but are non-optimal, and Point C is overcapacity with respect to transporting sediment.  227  0 Cl)  C B  Width  Figure 7.16. Solution curves used for a comparison of numerical models. The solution curves are schematic. The solution curves represent two different imposed sediment loads, with the upper curve corresponding to the higher imposed load. Points A and B are at the optima of their respective solution curves.  228  Non-Feasible  \\ \ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\  a .2  CD  /B\ \ s=S V  /C Feasible  /  A  Width  Figure 7.17. A solution curve used to interpret the behaviour of a meandering river that has been 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 straightened channel will be located at Points B or C, both of which are non-optimal.  229  CHAPTER 8  CONCLUSIONS AND RECOMMENDATIONS  8.1 SUMMARY  In this thesis an optimization model has been developed to predict the stable, 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 flowduration curve; the mean annual bedload, both total mass and 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 and bank stability to determine a channel geometry which is optimal as defined by  ,  which is the coefficient of sediment transport  efficiency (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,  230  however the degree of scatter associated with these models limited their application to quantitative 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 in this 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 bank sediment that was unaffected by bank vegetation. It has subsequently been extended to include the 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 published data of Andrews (1984) and Hey and Thome (1986). It was shown that the inclusion of the bank stability procedure significantly improves the model performance. The results of the influence of the bank vegetation on the channel stability are in good agreement with the results that Andrews and Hey and Thorne obtained from empirical regime analyses. The influence of the bank vegetation is to stabilise the bank sediment, allowing the banks to withstand higher shear stresses. This results in channels that are narrower, deeper, and less steep than their unvegetated counterparts. The change in W is the largest, followed by  Y, and the smallest  change is in S.  The analysis of channels with cohesive banks indicates that channels tend to be either bank height 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 be  231  bank 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 is does not influence the channel geometry. It is therefore necessary to determine which of the two possible constraints is active in order to assess the stability of the channel. Failure to do so can lead to an assessment of the channel stability based on the degenerate constraint which will lead 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 banks indicates that the bank vegetation has a strong effect on the value of with treed banks have higher values of  rcrit  Tcrir  (Fig 6.8). Channels  than channels with grassed banks. This results  indicates that the effect of bank vegetation on channels with cohesive banks is similar to the noncohesive case and that rivers with heavily vegetated banks would be narrower, deeper, and less steep than their unvegetated counterparts.  These results regarding the bank vegetation have important consequences for stream management as the removal of the bank vegetation can reduce çS or  zrjt  and destabilise the  channel. Published observations have shown that streamside logging can result in increased width and destabilisation of alluvial channels (Roberts and Church, 1986; Hartman and Scrivener, 1990). The optimization modelling could be used to determine the sensitivity of a channel to streamside logging.  232  8.1.2. Modelling Using The Full Flow-Duration Data  Previous optimization models such as Chang (1979, 1980) and White et aL (1982), as well as the simplified optimization models presented in Chapter 6 calculated the sediment transporting capacity of the channel based only on the bankfull rate. It was argued in Chapter 4 that this is inadequate 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 the time where Qbf is equalled or exceeded determines the total volume of sediment that is transported by the channel over one year. The final version of the optimization model presented in 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 considered and as a first approximation an infinitely wide flood plain is assumed. The depth of flow within the channel, and more importantly, the values of  vE,d  and  Thank  reach their maximum value at  Qbj; 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 simply 50 is considered to be a dependent variable. The size distribution of the bed surface and the D by ,  bedload transporting capacity of the channel are calculated using the Parker (1990) surfacebased 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 significant duration which is generally assumed to be one water year. Therefore the “equal mobility”  233  concept as applied herein is that the bed surface or armour layer forms so as to render all size fractions equally mobile over a period of one year. Therefore the grain size distribution of the mean annual load transported by a channel in equilibrium is exactly the same as the distribution of the imposed load. It is assumed that the grain size distribution of the mean annual imposed bedload is identical to the that of the subarmour sediment (Parker, 1990).  The definition of equal mobility developed in this thesis is significantly different from the original definition by Parker et a!. (1982a) who assumed that equal mobility was achieved for each flow. This is known to be incorrect from field observations (eg Fig. 4.11), and the original definition of equal mobility was acknowledged by Parker et a!. as a first-order approximation only.  8.1.4 Bankfull Discharge as a Dependent Variable  It was shown in Chapter 7 that there is an optimal value of Qbf for a given set of independent variables. 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 relatively insensitive to  Qbf,  i  is  and from this it would be expected that the observed values from  natural rivers would be randomly scattered across a wide range about the optimal value. This is an interesting result which deserves fhrther attention to determine its significance.  234  8.1.5 Verification of the Optimization Model Modelling results of Chang (1979, 1980) and White et at. (1982) showed reasonably good agreement between modelled and observed values of W and  7, although the degree of scatter  was 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 bank stability constraint. In addition Yang et a!. (1981) have shown that their optimization model yields the exponents of well-known empirical regime equations.  Furthermore good quantitative agreement was obtained between the modelling in Chapter 6 and Millar and Quick (1993a, b) for the effect of bank vegetation on channels with noncohesive channel banks, when compared to the empirical regime equations of Andrews (1984) and Hey and Thorne (1986).  In addition general qualitative agreement between the optimization modelling and the formulas of 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 been presented in this thesis. This theory was advanced beyond that which could be fully verified  within the scope of this thesis. A program for full verification of the optimization model is outlined in Section 8.2.6.  235  8.2 RECOMMENDATIONS FOR FUTURE WORK The optimization model presented herein is formulated using the most suitable equations that were available from the literature. Some modifications were made, however no experimental or field 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 are presented below.  8.2.1 Channel Form or Bar Roughness In 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) were shown to be a result of spurious correlation and therefore do not yield meaningful estimates of the form roughness. The approach adopted by Prestegaard (1983) is promising although no predictive equations have yet been developed.  The nature of the form roughness can be studied using an approach similar to that adopted by Prestegaard (1983). The total channel roughness,J can be determined from channel surveys at a location where good quality flow measurements have been obtained, preferably at a hydrometric station. The grain roughness,f’, can be calculated using Eqn (3.8) and the value of f”is equal to!  -  f’  from Eqn (3.7). Data obtained from detailed geomorphic mapping of the  channel, such as bar amplitude and spacing, vegetation etc, can be then used in an effort to develop empirical relations forf”.  236  8.2.2 Surface-Based Transport Relation Based on  t’bd  The Parker (1990) surface-based transport relation uses the total bed shear stress  Tbed,,  and not  the grain shear stress r ‘L,ed,, to calculate the transport rate. The use of the total bed shear stress and not the grain component is a result of Parker and Peterson (1980) who concluded that the form component of roughness and shear stress in gravel rivers is negligible, particularly for high in-bank flows. This was shown to be incorrect in Section 3.2, and furthermore it was shown in Section 4.2.4 that as little as 11% of the total shear stress can be assigned to the grain component (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 a  coefficient, but rather all of the relations which include the hiding and straining functions need to be recalculated.  8.2.3 Bank Stability  The results in Chapter 6 indicated that for both noncohesive and cohesive banks the bank vegetation exerts a strong influence on the bank stability, and hence the channel geometry. The effect of the bank vegetation on the bank stability is reduced to a single parameter which is ç for noncohesive bank sediment, or  rent  for cohesive bank sediment. In both cases the effect of  the bank vegetation was interpreted as an increase in  and  znjt  which resulted from the  binding and reinforcement of the sediment by the root masses. An alternative explanation is the physical shielding of the bank sediment by the vegetation which reduces the magnitude of acting on the bank sediment. Field investigation is necessary to determine the actual role of the 237  bank vegetation and how it relates to bank stability, and whether the effect is an increase in g or  rrjt,  or a decrease in  Tbank.  Furthermore it is necessary to develop field-based methods for  determining the insitu values of g and  rit  8.2.4 Secondary Currents The “strong” secondary currents that occur as a result of channel curvature have not been considered in this thesis. The “weak” secondary currents which are present in straight channels are implicitly accounted for in the empirical boundary shear stress equations of Flintham and Caning (1988) which were developed from experimental data.  One important effect of secondary currents is the redistribution of the boundary shear stress. In this thesis the bank and bed shear stresses are assumed to be uniform across their respective channel perimeters. However redistribution of the shear stress can have a large effect for example 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 the critical value for bed mobilisation. This channel will, based upon the mean value, transport a negligible volume of sediment. However if secondary currents redistribute the shear stress such that one half of the channel bed has a shear stress value about 1.5 times the critical value, and the other half is only about 0.5 times the critical value, significant sediment transport will occur across one half of the channel.  238  From the optimization model it is assumed that a reduction in the channel slope implies a decrease in the sediment transporting capacity of the channel. However this reduction in channel slope is presumed to be accompanied by increased sinuosity, resulting in stronger secondary currents which could conceivably increase the sediment transporting capacity of the channel.  The optimization model calculates the mean value of Vbank, and this value must be less than the shear stress required for fluvial erosion of the bank sediment in order to satisify the bankstability constraint. Locally however, the value of Tbank may exceed the value required for bank erosion, particularly in areas such as along the outer bank of meanders. Along the convex inner bank 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 is still considered to be in equilibrium, and “statistically stable”, if there is no net change in the mean channel geometry over a representative channel reach.  8.2.5 Formulation for Sand-Bed Rivers  The equations used in this thesis have been selected and developed for application to gravel-bed rivers, however the model can easily be reformulated for application to sand-bed rivers. For example the Einstein transport relation could be used in place of the modified Parker (1990) relation.  239  8.2.6 Model Verification Before the optimization model can be used for quantitative engineering studies to predict the impact of engineering structures or land-use practices on the channel geometry a fhll program of 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 approximate equilibrium. Existing data sets such as Kellerhals et a!. (1972) were found to lack certain key data. However these data sets could be updated with some field work to determine the values of the bank stability parameters, and review of the original records such as flow data, sediment analyses etc.  The values of the independent variables obtained from field surveys can be input into the model and a comparison made between the modelled and observed value of the dependent variables. 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Yang, C.T., 1976: “Minimum unit stream power and fluvial hydraulics.” I Hydr. Div., ASCE, 102 (7), 919-934.  Yang, C.T., and Song, C.C.S., 1979: “Theory of minimum rate of energy dissipation.” I Hyd.r. Div., ASCE, 105 (7), 769-784.  Yang, C.T., Song, C.C.S., and Woldenberg, M.J., 1981: “Hydraulic geometry and minimum rate of energy dissipation.” Water Res. Res., 17(4), 1014-1018.  259  Yang, C.T., and Song, C.C.S., 1986: “Theory of minimum energy and energy dissipation rate.” In Encyclopedia of fluid mechanics, N.P. Cheremisinoff (Ed.), Vol. 1, Chapt. 11, Gulf Publishing Co., Houston, Texas, 3 53-399.  260  APPENDIX A  BANKFULL: FIXED SLOPE MODEL  The Bankfhll:Fixed Slope model is the simplest version of the optimization model presented in this thesis. 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 the optimal geometry in a data file named fixslope.out. The data file can contain the data for more than one channel. The data for each channel must be input as follows: Qbf  S  ksbd  ksbk  50 d  50 D  ‘isis  D5Obk  for channels with noncohesive banks, and: Qbf  S  ksbd  ksbk  50 d  50 D  rt  C’  Tcrit  for channels with cohesive banks.  The data relating to each channel must start on a new line. The data values must be separated by at least one space, although the number of spaces is not important. An example of an input dat file is presented below: 50 100  0.003 0.003  0.1 0.1  0.1 0.1  0.025 0.025  0.075 0.075  0.075 0.05  261  40 40  This file contains the input data for two channels with noncohesive banks. The user is required to input the bank sediment type (n for noncohesive or c for cohesive) as prompted. The output data will written to a data file named fixslope.out in the following order: W  H  17*  S  0  The output file resulting from the example input file is: * W H S theta 18.0 1.40 1.21 0.0030 30.0 37.7 1.28 1.17 0.0030 21.7  The Bankfiull:Fixed Slope model was developed principally for illustrative purposes. It can be used in a field investigation to estimate or ‘calibrate’ the bank stability parameters that may be difficult to measure. For example with noncohesive bank sediment the value of  Ø.,  may be  difficult to estimate in the field. The measured independent variables can be input into the model together with trial values of  until the modelled width is equal to the observed channel  width. This procedure was used in Chapter 6 to demonstrate the relationship between bank vegetation. The value of  and  which gives the best agreement between the modelled and  observed 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  0  DECLARE SUB bankstabilitycohesive 0 DECLARE SUB optimum 0 DECLARE SUB bankstabilitynoncohesive  0  DECLARE SUB ebrown 0 DECLARE FUNCTION bankshear! 0 DECLARE FUNCTION bedshear! 0 DECLARE FUNCTION shearforce! 0 DECLARE SUB continuity 0 DECLARE FUNCTION velocity! 0 DECLARE FUNCTION hydrad! 0 DECLARE FUNCTION meandepth! 0 DECLARE FUNCTION area? 0 DECLARE FUNCTION surfwidth! 0 DECLARE FUNCTION depth? 0 I*********************************************************************************  program to calculate curves to demonstrate the optimum channel geometiy for a given set of independent variables ‘bankfull: fixed slope model for channels with ‘both cohesive and noncohesive bank sediment  ‘Global Variables gamma? unit weight of water viscosity of water viscosity?=kinematic sediment specific gravity of ‘ss! slope slope?=channel water of ‘density!=density pbank?—bank perimeter ‘pbed!=bed perimeter armour grain diameter (radians) dd5O?=median angle ‘theta!=bank grain diameter of bank sediment d5Obank?=median diameter grain ‘d50!median subarmour bedload transport rate ‘neta!=coefficient of ‘gbcalc!=valculated ‘discharge!=bankfull discharge efficiency ‘phipnme?=friction angle of bank sediment (radians) ‘phidegrees!=friction angle of bank sediment (degrees) ksbank!=measure of bank roughness ‘ksbed?=measure of bed roughness Wointerger counter sediment ‘banktype$=type of bank bank sediment cohesive ‘gammat!=unit weight of for cohesive sediment number ‘stabnum’=stability taucrit?=critical shear stress cohesion ‘cohesion!=soil ‘gravity!= gravitational constant  COMMON 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 #1 OPEN “flxslope.out” FOR OUTPUT AS #2  263  ‘SET VALUES OF INDEPENDENT CONSTANTS pi! = 3.14159 gravity!= 9.81 gamma! = 9810 viscosity! = .000001 density! = 1000 ss! = 2.65 CLS PRINT PRINT PRINT “BANKFULL:FIXED SLOPE OPTIMIZATION MODEL” PRINT : INPUT “Hit <ENTER> to Continue “,dummy$  =0 PRINT #2,” W  H  *  S theta”  CLS 10 PRINT PRINT PRINT PRiNT : INPUT “INPUT BANK SEDIMENT TYPE (n/c)  “;  banktype$  iF banktype$ = “c” OR banktype$ = “C” THEN banktype$ = “cohesive” ELSEIF bankte$ = “N” OR banktype$ = “n” THEN banktype$ = “noncohesive” ELSE BEEP PRINT “Bank Type Unknown” GOTO 10 END IF CLS  DO WHILE NOT EOF(1) PRINT PRINT =1% + 1 ‘counter PRiNT “Channel Number “; f’% iNPUT #1, discharge!, slope!, ksbed!, ksbank!, d50!, dd5O! ‘Input the independent variables from data file IF banktype$ “noncohesive” THEN ‘input bank stability parameters INPUT #1, d50bank!, phidegrees! ELSE ‘banktype$=”cohesive” INPUT #1, gamxnat!, phidegrees!, cohesion!, taucrit! END IF  264  phiprime!  =  phidegrees!  *  2  *  pi! / 360 ‘convert friction angle from degrees to radians  CALL optimum ‘calculates the optimal geometry PRINT * PRINT” W H S theta” PRINT USiNG “###.# ##.## ##.## #.#### pi! PRINT #2, USING / pi!  “###.#  ##.#“;  surfwidth!; depth!; meandepth!; slope!; theta!  *  360/2 /  ##.## ##.## #.#### ##.#“; surfwidth!; depth!; meandepth!; slope!; theta!  *  LOOP ‘loops until end of data file CLOSE #1 CLOSE #2  END  FUNCTION area ‘function to calculate the cross-sectional area area!  =  .5  *  (pbed! + surfwidth!)  *  depth!  END FUNCTION  FUNCTION bankshear ‘calculates the bank shear stress bankshear!  =  gamma!  *  depth!  *  slope!  *  shearforce!  *  ((surfwidth!  END FUNCTION  SUB bankstabilitycohesive ‘satisfies bank stability constraint for cohesive banks  heightcond$ = “unknown” shearcond$ = “unknown” bankcond$ = “unknown” thetamax! =90*2 *pi! /360 thetanün! =0 ‘initialise search and convergence criteria  ‘First test if vertical bank is stable wrt bank height theta! = thetamax! CALL continuity ‘satisfy continuity  265  +  pbed!)  *  SIN(theta!) / (4  *  depth!))  360 / 2  CALL stabcurve  ‘catculate stability number criticalheight! = stabnuml * cohesionl I gaxnmat’ calculate the critical height stability 1!  = depth / criticaiheight! ‘calculates stability criteria wit bank height ‘if stability 1 I <= criticaiheight then bank is stable ‘with respect to bank height  IF stabilityll <= 1.001 THEN heightcond$ = “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” 1 theta  =  (thetamax? + thetamin’) /2 ‘calculate midpoint of range for ‘bisectrix convergence scheme  CALL continuity ‘satisfy continuity constraint CALL stabcurve ‘calculate stability number criticaiheight! = stabnum! * cohesion! / gaimnat! stabilityll = depth / criticaiheight’ ‘calculates critical height and stabilty number ‘for the bank height constraint  IF stabilityl! >= 1.001 THEN heightcond$ = “unstable” thetamax! theta? ELSEIF stabilityll <= .999 THEN heightcond$ = “understable” thetamin! = theta! ELSE heightcond$ = ‘just stable” ‘Bank Height = Critical Height END IF IF (thetamax! thetamin’) / theta! <.001 THEN heightcond$ = ‘Just stable” stabilityl! = 1 ‘Secondary Convergence Criterion in case of ‘convergence problems due to numerical scheme END IF -  LOOP 266  END IF  bank height constraint now satisified ‘now assess the bank shear constraint thetaniin?=0 thetamax! = theta! ‘thetamax is reset to maximum bankangle which satisfied the ‘bank height constraint above stability2! = bankshear! / taucrit satbility criterion for bank shear constraint IF stability2! < 1.001 THEN ‘bank shear constraint is satisifled shearcond$ = “stable” bankcond$ = “stable” ELSE ‘must reduce theta! DO UNTIL shearcond$ theta!  =  =  “just stable”  (thetamax! + thetaniin!) /2 ‘calculate mid point  IFtheta! < 10*2*3.14159/36OTHEN ‘bank angles less than 10 degrees ‘nominally assumed to be unstable shearcond$ = “unstable” bankcond$ = “unstable” EXIT DO END IF CALL continuity ‘satisfy continuity stability2! = bankshear! / taucrit! ‘calculate stability criterion for bank shear constraint  IF stability2! >= 1.001 THEN ‘bankshear > taucrit shearcond$ = “unstable” thetamax! = theta! ELSEIF stability2! <= .999 THEN shearcond$ = “understable” thetamin! theta! ELSE shearcond$ = “just stable” bankcond$ = “stable” 267  END IF  IF (thetaxnax! thetanun!) / theta! <.001 THEN shearcond$ ‘just stable” bankcond$ = “stable” stability2? = 1 END IF -  LOOP END IF END SUB  SUB bankstabilitynoncohesive ‘calculates theta where banks just stable  bankcond$ = “unknown” thetamax! = phiprime! thetamin! =0 ‘initialise search bounds and convergence criteria DO UNTIL bankcond$ = ‘just stable” theta! (thetaniax! + thetaniin!) / 2 ‘determine midpoint of search range IF theta! <5 * 2 * 3.14159/360 THEN bankcond$ = “unstable” EXiT DO ‘if theta is less than 5 degrees the channel is assumed ‘to be unstable END IF CALL continuity ‘satisfies continuity constraint stability! = (bankshear! / (gamma! SIN(phiprime!) A 2) A .5) ‘calculates stability criterion  *  (ss!  -  1)  *  d5obank!)) I (.048  IF stability! >= 1.001 THEN bankcond$ = “unstable” thetamax! = theta! ELSEIF stability! <= .999 THEN bankcond$ = “stable” thetamin! = theta! ELSE bankcond$ = ‘just stable” ‘primaiy convergence criterion END IF  268  *  TAN(phiprime!)  *  (1  -  SIN(theta!) A 2 /  IF (thetamax! thetamin!) I theta! <.001 THEN bankcond$ = ‘just stable” stability! = 1 ‘secondaiy convergence criterion END IF LOOP -  END SUB  FUNCTION bedshear ‘calculates the bed shear stress  gamma!  bedshear!  *  depth!  *  slope!  *  (1  -  shearforcel)  *  ((surfwidth! / (2  *  pbed!)  +  .5))  END FUNCTION  SUB continuity ‘varies pbank! for trial values of Pbed!, slope!, and theta! to ‘satisfy the contiuity constraint pbankcond$ = “unknown” errorcalc! = 1000 minpbank! =0 maxpbank! =20 * discharge! A •35 ‘initialise search and convergence criteria  DO UNTIL pbankcond$ pbank!  =  “OK”  (minpbank! + maxpbank!) /2 ‘calculate midpoint  =  errorcalcl = (area * velocity / discharge!) ‘calculate the normalised error  IF errorcaic! > 1.001 THEN pbankcond$ “too large” maxpbank! = pbank! ELSEIF errorcalc! <.999 THEN pbankcond$ = “too small” minpbank! = pbank! ELSE  pbankcond$  =  “OK”  END IF  IF (maxpbank! minpbank!) / pbank! <.0001 AND pbankcond$ ‘resets maxpbank! if too small maxpbank! =2 * maxpbank! END IF -  269  =  “too small” THEN  LOOP END SUB  FUNCTION depth ‘function to calculate flow  depth!  .5  =  *  SIN(theta?)  depth of a trapezoidal channel  *  pbank!  END FUNCTION  SUB ebrown ‘calculate the sed trans capacity using em-brown formula dimlessbedshear!  =  bedshear! / (gamma!  *  (ss!  -  1)  *  d50!)  IF dinflessbedshear! <=0 THEN dimlessbedload! =0 ELSEIF dimlessbedshear! <.093 THEN dimlessbedload! = 2.15 * EXP(-.391 / dimlessbedshear!) ELSE dimlessbedload! =40 * dimlessbedshear! “3 END IF fi! = (2 / 3 + 36 *(ss!_1))) gbcalc!  =  pbed!  viscosity? “2/ (gravity!  *  *  fi!  *  ss?  *  density!  *  *  ((ss!  d50! “3  -  1)  *  *  (ss?  gravity!  END SUB  FUNCTION hydrad! ‘function to calculate  hydrad!  =  area!  the hydraulic radius  / (pbed?  +  pbank!)  END FUNCTION  FUNCTION meandepth ‘function to calculate the mean depth meandepth!  =  area  / surfwidth  END FUNCTION  SUB optimum ‘determines the optimal geometzy optcond$  = “unknown” ‘initialises optimality test condition  270  *  -  1))) “.5 -(36  d50!  A  3)  A  5  *  *  viscosity? “2/ (gravity!  dimlessbedload!  *  d.50? “3  lowerpbed! =0 upperpbed? = 10 * discharge! “.5 ‘set bounds for Pbed based on Regime Eqns ‘typical optimal value of Pbed is 2 to 5 * discharge” .5 minpbed! = lowerpbed! maxpbed! = upperpbed! PRINT DO WHILE optcond$ pbed!  <>  “optimum”  (minpbed! + maxpbed!) /2 ‘calculate mid point of search range  =  IF pbed!  .95  *  upperpbed! THEN upperpbed!2*upperpbed! maxpbed! = upperpbed! ‘reset upper bound of search END IF >  PRINT “Assessing Trial Bed Perimeter PRiNT USiNG “####.##“; pbed’; PRiNT “ m “; pbed! = pbed! * .975  “;  ‘calculate backwards difference value of Pbed IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE ‘banktype$ = “cohesive” CALL bankstabilitycohesive ‘solve bank stability constraint for cohesive banks END IF IF bankcond$ <> “unstable” THEN ‘test conditions indicate that stable channel geometry ‘has been determined and the bedload constraint ‘has been satisified CALL ebrown ‘calculate sdiment tarnsporting capacity of the channel netal!  pbed!  discharge! * slope!) ‘evaluate neta for backward difference point  =  gbcalc! / (density!  *  pbed! 1.975 * 1.025 ‘calculate pbed for forward duff  IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE ‘banlctype$ = “cohesive” CALL bankstabilitycohesive ‘solve bank stability constraint for cohesive banks END IF CALL ebrown 271  ‘calculate sdiment tarnsporting capacity of the channel  neta2!  gbcalcl / (density’ * discharge! * slope!) ‘calculates neta for forward difference value  =  pbed! =pbed! / 1.025 ‘reset pbed to midpoint value dnetabydpbed! = (neta2! netal!) I (.05 * pbed!) ‘calculate first derivative by finite difference ‘to assess optimum condition -  IF dnetabydpbed? <0 THEN ‘Trial Pbed is too de maxpbed! = pbed’ ‘Reduce upper bound of Pbed optcond$ = “too wide” ELSEIF dnetabydpbed! >0 THEN ‘Trial Pbed is too narrow minpbed! = pbed! ‘Increase lower bound of Pbed optcond$ = “too narrow” ELSE optcond$ = “optimum” ‘Optimum Achieved END IF IF (maxpbed!  -  minpbed!) I pbed! <.001 THEN optcond$ “optimum” ‘Convergence attained. Second optimality criterion  ELSE ‘trial Pbed too small for stable geometiy: Increase minimum Pbed pbed! =pbed!/.975 ‘reset from backward difference to correct value minpbed! = pbed! ‘set lower limit at current trial value ‘(as the optimum value must be greater) optcond$ = “too narrow” END IF PRiNT optcond$ LOOP  ‘Evaluate geometiy at exact optimal Pbed ‘(not at forward or backward difference value) IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE banktype$ = “cohesive” CALL bankstabilitycohesive ‘solve bank stability constraint for cohesive banks END IF CALL ebrown ‘calculate sdiment tarnsporting capacity of the channel neta!  =  (pbed!  *  gbcalc!) / (density!  *  discharge!  *  slope!) 272  ‘evaluate objective function  END SUB ‘*********************************************************************************  FUNCTION shearforce ‘calculates the proportion of shear force on the banks shearforce!  =  1.766  *  (pbed! I pbank!  +  1.5)  A  -1.4026  END FUNCTION ‘*********************************************************************************  SUB stabcurve ‘stability curves are from Figure 5.3 thetadegrees! =theta! /2/3.14159  *  360  IF phidegrees! <20 THEN ‘use phiprime = 15 degree stability curve from Figure 5.3 ‘from Taylor (1948) SELECT CASE thetadegrees! CASE 48.65 TO 90 stabnum! = -.141  *  thetadegrees!  +  17.46  CASE 29.36 TO 48.6499 stabnum! = -.533 * thetadegrees!  +  36.53  CASE 20.12 TO 29.3599 stabnum! = -3.07 * thetadegrees! CASE 15.001 TO 20.11999 stabnum! = -189.6  +  111  thetadegrees!  *  CASEIS<= 15 stabnum!  =  1000000  END SELECT ELSE ‘use phiprime =25 degrees curve from Figure 5.3 ‘After Taylor, (1948) SELECT CASE thetadegrees! CASE 58.62 TO 90 stabnum! = -.212 * thetadegreesi  +  CASE 39.11 TO 58.61999 stabnum! = -.8975 * thetadegrees! CASE 25 TO 39. 10999 stabnuml = -6.97 * thetadegrees!  24.86  +  +  65.04  302.5 273  +  3844  CASE IS <=25 stabnuml  =  1000000  END SELECT END IF END SUB ‘*********************************************************************************  FUNCTION surfwidth  ‘function to calculate the surface width surfwidth!  =  pbed!  +  COS(theta?)  *  pbank?  END FUNCTION  FUNCTION velocity! ‘calculates the mean velocity rhbank! = bankshear! I (gamma! * slope!) rhbed! = bedshear! I (gamma! * slope!) Thank! = (2.03 * LOG(12.2 * rhbank! I ksbank!) I LOG(10)) A -2 Ibed! = (2.03 * LOG(12.2 * rhbed! / ksbed!) I LOG(10)) A -2  ifactor!  =  velocity!  (ibed!  =  (8  *  *  pbed! I (pbed!  gravity!  *  +  hydrad!  pbank!) + Ibank! *  slope!)  A  5  *  *  pbankl / (pbank! + pbed!))  ifactor!  END FUNCTION  274  A  APPENDIX B  BANKFULL: VARIABLE SLOPE MODEL  The Bankfull:Variable Slope differs from the Fixed Slope model in that the channel slope is now 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 output to a data file named varslope.out. The data file can contain data for more tahn one channel. The data for each channel must be input as follows: Qbf  Gbf  ksd  ksbk  50 d  50 D  D5Obk  s  for channels with noncohesive banks, and: Qbf  ksbd  Gbf  k,bk  50 d  50 D  rt  c’  rcrit  for channels with cohesive banks.  The data relating to each channel must start on a new line. The data values must be separated by at least one space, although the number of spaces is not important. An example of an input dat file is presented below: 50 100  5 20  0.1 0.1  0.1 0.025 0.075 0.075 0.1 0.025 0.075 0.05  275  40 40  This file contains the input data for two channels with noncohesive banks. The user is required to input the bank sediment type (n for noncohesive or c for cohesive) as prompted. The output data will written to a data file named varslope.out in the following order:  W  H  17*  S  6  The output file resulting from the example input file is: W 16.5 35.3  * theta H S 1.66 1.39 0.0023 32.0 1.39 1.26 0.0027 22.2  The Bankfiill:Variable Slope model was developed principally for illustrative purposes. The source code for the Bankfull:Variable Slope Model is presented below.  276  BANKFULL:VAR1ABLE SLOPE MODEL FOR BOTH COHESIVE AND NONCOHESIVE BANK SEDIMENT ‘(VARSLOPE.BAS) DECLARE SUB stabcurve 0 DECLARE SUB bankstabilitycohesive 0 DECLARE SUB bedload2 0 DECLARE SUB optimum 0 DECLARE SUB bankstabilitynoncohesive DECLARE SUB bedload 0 DECLARE SUB ebrown 0 DECLARE FUNCTION bankshear! 0 DECLARE FUNCTION bedshear? 0 DECLARE FUNCTION shearforce? 0 DECLARE SUB continuity 0 DECLARE FUNCTION velocity! 0 DECLARE FUNCTION hydrad! 0 DECLARE FUNCTION meandepth! 0 DECLARE FUNCTION area? 0 DECLARE FUNCTION surfwidth! 0 DECLARE FUNCTION depth! ()  0  program to calculate curves to demonstrate the optimum channel geometiy for a given set of independent variables ‘bankfull: variable slope model for channels with ‘both cohesive and noncohesive bank sediment  ‘Global Variables gamma?= unit weight of water ‘gravity!= gravitational constant viscosity!=kinematic viscosity of water = specific gravity of sediment 1 ‘ss slope!=channel slope ‘density!=density of water pbank!=bank perimeter ‘pbedl=bed perimeter dd5O!=median armour grain diameter ‘theta!=bank angle (radians) ‘d50!inedian subarinour grain diameter d5Obank!=median grain diameter of bank sediment gbload!=imposed bedload transport rate ‘discharge!=bankfull discharge ‘neta!coefficient of efficiency ‘gbload!=calculated bedload transport rate ‘phiprime!=friction angle of bank sediment (radians) ‘phidegrees!=friction angle of bank sediment (degrees) ksbank!measure of bank roughness ‘ksbcd!=measure of bed roughness Wo=interger counter ‘banktype$=type of bank sediment ‘gammat!=unit weight of cohesive bank sediment ‘stabnuxn!=stability number for cohesive sediment taucrit?=cntical shear stress ‘cohesion!=soil cohesion  COMMON SHARED gravity!, gamma!, ss!, viscosity?, density? COMMON SHARED slope!, pbed?, pbank!, theta!, dd5O!, d5Obank! COMMON SHARED discharge!, gbcalc!, gbload!, phiprime? COMMON SHARED d50!, bankcond$, pi! COMMON SHARED ksbed!, ksbank!, neta!, slopecond$, banktype$, 1% COMN’ION SHARED ganimat!, stabnum!, cohesion!, taucrit?, phidegrees! 277  OPEN “varslope.dat” FOR iNPUT AS #1 OPEN “varslope.out” FOR OUTPUT AS #2 ‘SET VALUES OF INDEPENDENT CONSTANTS pi! = 3.14159 gravity! = 9.81 gamma! = 9810 viscosity! = .000001 density! = 1000 ss! =2.65 CLS PRINT PRINT PRINT “BANKFULL: VARIABLE SLOPE OPTIMIZATION MODEL” PRINT : INPUT “Hit <ENTER> to Continue “, dummy$  =0 S theta” CLS 10 PRiNT PRINT PRINT PRINT : INPUT “INPUT BANK SEDIMENT TYPE (a/c) “;banktype$ PRINT #2,” W  H  *  = “c” OR banktype$ = “C” THEN banktype$ = “cohesive” ELSEIF banktype$ = “N” OR banktype$ = “n” THEN banktype$ = “noncohesive” ELSE BEEP  IF banktype$  PRINT “Bank Type Unkno”  GOTO 10 END IF  CLS DO WHILE NOT EOF(l) PRINT PRINT I% =1% + 1 ‘counter PRiNT “Channel Number “; We INPUT #1, discharge!, gbload!, ksbed!, ksbank!, d50!, dd5O! ‘Input the independent variables from data file  IF banktype$ = “noncohesive” THEN ‘input bank stability parameters INPUT #1, d50bank!, phidegrees! ELSE ‘banktypeS=”cohesive” iNPUT #1, ganunat!, phidegrees!, cohesion!, taucrit! END IF  278  phiprimel  =  phidegrees!  *  2  *  pit / 360 ‘convert friction angle from degrees to radians  CALL optimum ‘calculates the optimal geometiy PRiNT * PRINT” W H S theta” PRiNT USiNG “##*.# #1.## #.## #.## #.#“; surfwidth!; depth!; meandepth!; slope!; theta!  *  360 / 2 /  pi! ##.## #.#### ##.#“; surfwidth!; depth!; meandepth!; slope!; theta!  PRINT #2, USING “###.#  *  360 /2  / pi! ‘prints out the values of the dependent variables LOOP ‘loops until end of data file CLOSE #1 CLOSE #2  END FUNCTION area ‘function to calculate the cross-sectional area area!  =  .5  *  (pbed! + surfwidth!)  *  depth!  END FUNCTION FUNCTION bankshear  ‘calculates the bank shear stress bankshear!  =  gamma!  *  depth!  *  slope!  *  shear.force!  *  ((surfwidth!  END FUNCTION SUB bankstabilitycohesive ‘satisfies bank stability constraint for cohesive banks  heightcond$ = “unknown” shearcondS = “unknown” bankcond$ = “unknown” thetaniax! =90*2*pi!/360 thetamin! =0 ‘imtialise search and convergence criteria  ‘First test if vertical bank is stable wrt bank height theta! = thetamax! CALL continuity ‘satisfy continuity CALL stabcurve ‘catculate stability number  279  +  pbed!)  *  SIN(theta!) / (4  *  depth!))  stabnum? * cohesion! I ganunat! calculate the critical height  criticalheight!  =  stability!! = depth I criticalheight! ‘calculates stability criteria wrt bank height ‘if stabilityl! <= criticaiheight then bank is stable ‘with respect to bank height IF stabilityl! < 1.001 THEN heightcond$ = “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 scheme  CALL continuity ‘satisfy continuity constraint CALL stabcurve  ‘calculate stability number criticaiheight! = stabnum! * cohesion! / ganunat! stability 1! = depth / criticaiheight! ‘calculates critical height and stabilty number Tor the bank height constraint  IF stabilityl! >= 1.001 THEN heightcond$ = “unstable” thetamax! =theta! ELSEIF stabilityl! < .999 THEN heightcond$ = “understable” thetamin! = theta! ELSE heightcond$ = “just stable” ‘Bank Height = Critical Height END IF IF (thetamax! thetamin!) / theta! <.001 THEN heightcond$ = “just stable” stability!! = 1 ‘Secondary Convergence Criterion in case of ‘convergence problems due to numerical scheme END IF -  LOOP END IF  280  ‘bank height constraint now satisified ‘now assess the bank shear constraint thetamin! =0 thetaniax! = theta! ‘thetamax is reset to maximum bankangle which satisfied the ‘bank height constraint above stability2! = bankshear! / taucrit! ‘satbility criterion for bank shear constraint IF stability2! <1.001 THEN ‘bank shear constraint is satisified shearcond$ = “stable” bankcond$ = “stable” ELSE ‘must reduce theta! DO UNTIL shearcond$ theta?  =  =  “just stable”  (thetamax! + thetamin!) /2 ‘calculate mid point  IFtheta! <10*2*3.14159/360 THEN  ‘bank angles less than 10 degrees ‘nominally assumed to be unstable shearcond$ = “unstable” bankcond$ = “unstable” EXIT DO END IF CALL continuity  ‘satisfy continuity stability2! = bankshear! / taucrit! ‘calculate stability criterion for bank shear constraint  IF stability2! >= 1.001 THEN ‘bankshear > taucrit shearcondS = “unstable” thetamax! =theta! ELSEIF stability2! <= .999 THEN shearcond$ = “understable” thetainin! =theta! ELSE shearcond$ = “just stable” bankcond$ = “stable” END IF IF (thetamax! thetamin!) / theta! <.001 THEN -  281  shearcond$ = “just stable” bankcond$ = “stable” stability2! = 1 END IF  LOOP END IF END SUB  SUB bankstabilitynoncohesive ‘calculates theta where banks just stable bankcond$ = “unknown” thetamax! = phiprime’ thetamin! =0 ‘initialise search bounds and convergence criteria  DO UNTIL bankcond$ = ‘just stable” theta! = (thetamax’ ÷ thetamin!) /2 ‘determine midpoint of search range * 2 * 3.14159/360 THEN bankcond$ = “unstable” EXIT DO ‘if theta is less than 5 degrees the channel is assumed ‘to be unstable END IF  lFtheta! <5  CALL continuity ‘satisfies continuity constraint stability! = (bankshear! / (gamma! S1N(phiprime!) A 2) A .5) ‘calculates stability criterion  *  (ss!  -  1)  *  d50bank!)) / (.048  IF stability! >= 1.00 1 THEN bankcond$ = “unstable” thetamax! =theta! < .999 THEN bankcond$ = “stable” thetamin! = theta!  ELSEIF stability!  ELSE bankcond$ = “just stable” ‘primary convergence criterion ENDIF  IF (thetamax! thetainin!) I theta! <.001 THEN bankcond$ = ‘just stable” stability! = 1 -  282  *  TAN(phiprime!)  *  (1 SIN(theta!) A 2/ -  ‘secondary convergence criterion END IF LOOP END SUB  SUB bedload ‘satisfies the bedload constraint using the Einstein Brown sediment transport relation gbcalc!=0 minslope’ =0 maxslope! = .05 ‘set bound of slope search slopecond$ “unknown” ‘initialise slope condition DO WHILE slopecond$ <> “satisfied” slope? = (minslope! + maxslope!) I 2 ‘calculate mid point of range for bisectrix convergence scheme IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE ‘banktype$ = “cohesive” CALL bankstabilitycohesive ‘solve bank stability constraint for cohesive banks END IF  IF bankcond$ <> “unstable” THEN ‘bank stability is satisfied CALL ebrown ‘calculate sediment transporting capacity of trial channel IF gbcalc! <.999 * gbload! THEN ‘compares calculated to imposed sediment load slopecond$ = “too flat” minslope! = slope! ELSEIF gbcalc!> 1.001 * gbload? THEN slopecondS “too steep” maxslope! = slope! ELSE slopecond$ = “satisfied” ‘primary convergence criterion ‘sediment transporting capacity of the channel ‘equals the imposed load bedload constraint satisfied END IF IF (maxslope! minslope!) / slope! <.0001 THEN IF gbcalc! > .99 * gbload! THEN slopecondS = “satisfied” ‘secondary convergence criterion -  283  ‘when bounds of slope very small: convergence attained ‘occasionally a problem with convergence due to numerical ‘approximations ELSE slopecond$ = “not satisfied” EXIT DO ‘indicates that trial Pbed value is too narrow END IF END IF ELSE maxslope! = slope! ‘trial slope too steep: reduce upper bound slopecond$ = “unstable” END IF LOOP END SUB SUB 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 difference minslope! = 9 * slope! maxslope! = 1.1 * slope! ‘set bound of slope search slopecond$ = “unknown” ‘initialise slope condition DO WHILE slopecond$ <> “satisfied” slope! = (minslope! + maxslope!) /2 ‘calculate mid point of range for bisectrix convergence scheme IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE banktype$ = “cohesive” CALL bankstabilitycohesive ‘solve bank stability constraint for cohesive banks END IF IF bankcond$ <> “unstable” THEN ‘if bank stability is satisfied CALL ebrown ‘calculate sediment transporting capacity of trial channel IF gbcalc! <.999 * gbload! THEN ‘compares calculated to imposed sediment load slopecond$ = “too flat” minslope! = slope! ELSEIF gbcalc!> 1.001 * gbload! THEN 284  slopecondS  maxslope?  =  =  “too steep”  slope!  ELSE slopecond$ = “satisfied” ‘sediment transporting capacity of the channel ‘equals the imposed load : bedload constraint satisfied END IF IF (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 ‘approximations -  ELSE ‘if bank stability constraint not satisfied maxslope! = slope! ‘trial slope too steep: reduce upper bound slopecond$ = “unstable” END IF LOOP END SUB FUNCTION bedshear ‘calculates the bed shear stress bedshear!  =  gamma!  *  depth!  *  slope!  *  (1 shearforce!) -  *  ((surfwidth! / (2  END FUNCTION SUB continuity ‘varies pbank! for trial values of Pbed!, slope!, and theta! to ‘satisfy the contiuity constraint pbankcond$ = “unknown” errorcalci = 1000 minpbank! 0 maxpbank! =20 * discharge! “.35 ‘initialise search and convergence criteria  DO UNTIL pbankcond$ pbank’  =  “OK”  (minpbank! + maxpbank!) /2 ‘calculate midpoint  =  errorcalc! = (area * velocity / discharge!) ‘calculate the normalised error IF errorcalc! > 1.001 THEN pbankcond$ = “too large” 285  *  pbed!)  +  .5))  maxpbank!  =  pbank!  ELSEIF errorcaic! <.999 THEN pbankcond$ = “too small” minpbank! = pbank! ELSE pbankcond$  =  “OK”  END IF IF (maxpbankl minpbank’) / pbank? <.0001 AND pbankcond$ ‘resets maxpbank’ if too small maxpbank? =2* maxpbank! END IF -  “too small” THEN  LOOP END SUB FUNCTION depth ‘function to calculate flow depth of a trapezoidal channel depth!  =  .5  *  STN(theta!)  *  pbank’  END FUNCTION SUB ebrown ‘calculate the sed trans capacity using em-brown formula dimlessbedshear!  =  bedshear! / (gamma!  *  (ss!  -  1)  *  d50t)  IF dimlessbedshear! <=0 THEN dimlessbedload? = 0 ELSEIF dimlessbedshear! <.093 THEN dimlessbedload! = 2.15 * EXP(-.391 / dimlessbedshear’) ELSE dimlessbedload! = 40 * dimlessbedshear! “ 3 END IF fi! = (2/3 + 36 *(ss!_1)))  *  viscosity!  gbcalc! =pbed! *fl!  *!  A  2 / (gravity!  *  *deusity! *((ssi  d50’  -  A  3  *  (ss!  1)))  1)*gravity! *d5o!  END SUB FUNCTION hydrad! ‘function to calculate the hydraulic radius hydrad!  -  area! / (pbed! + pbank!)  END FUNCTION FUNCTION meandepth ‘function to calculate the mean depth 286  A  5 -(36  A3)A.5  *  viscosity!  A  2 / (gravity!  *dje55dicadf  *  dSO?  A  3  meandepth!  =  area / surfwidth  END FUNCTION  SUB optimum ‘determines the optimal geometry optcond$ = “unknown” ‘initialises optimality test condition lowerpbedl =0 upperpbedl = 10 * discharge? A ‘set bounds for Pbed based on Regime Eqns ‘typical optimal value of Pbed is 2 to 5 * discharge A minpbed! = lowerpbed? maxpbed! = upperpbedl PRINT PRINT  DO WHILE optcond$ pbed?  <>  “optimum”  (minpbedl + niaxpbed!) /2 ‘calculate mid point of search range  =  IFpbed’  *  upperpbed! TI-lEN upperpbedl =2 * upperpbed? maxpbed! = upperpbed? ‘reset upper bound if necessary  >  .95  END IF PRiNT “Assessing Trial Bed Perimeter “; PRINT USING “#####“; pbed?; PRINT “ m pbed’ =pbed? * .975 ‘calculate backwards difference value of Pbed  CALL bedload ‘satisfies continuity, bank stability and bedload ‘constraints for the trail Pbed value  IF slopecond$ = “satisfied” AND bankcond$ <> “unstable” THEN ‘test conditions indicate that stable channel geometry ‘has been determined and the bedload constraint ‘has been satisified netal!  gbload! / (density! * discharge! * slope!) ‘evaluate neta for backward difference point  =  pbed! =pbed! / .975 * 1.025 ‘calculate pbed for forward duff  287  CALL 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 computations  neta2  pbed’  gbload! / (density! * discharge! * slope!) ‘calculates neta for forward difference value  =  pbed! / 1.025 ‘reset pbed to midpoint value  dnetabydpbed! = (neta2! netal!) / (.05 * pbed!) ‘calculate first derivative by finite difference ‘to assess optimum condition -  IF dnetabydpbed! <0 THEN ‘Trial Pbed is too wide ‘Reduce upper bound of Pbed maxpbed! = pbed! wide” optcond$ = “too ELSETF dnetabydpbed! >0 THEN ‘Thai Pbed is too narrow ‘Increase lower bound of Pbed t = pbed! minpbed optcond$ = “too narrow” ELSE ‘Optimum Achieved optcond$ = “optimum” END IF  IF (maxpbed! minpbed!) / pbed! <.001 THEN optcond$ = “optimum” ‘Convergence attained. Second optimality criterion -  ELSE ‘tnal Pbed too small for stable geomeuy: Increase minimum Pbed pbed! I .975 ‘reset from backward difference to correct value minpbed! =pbed! ‘set lower limit at current trial value ‘(as the optimum value must be greater) optcond$ = “too narrow”  pbed!  =  END IF PRINT optcond$  LOOP  ‘Evaluate geometiy at exact optimal Pbed ‘(not at forward or backward difference value) IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE ‘banktype$ = “cohesive” CALL bankstabilitycohesive ‘solve bank stability constraint for cohesive banks END IF 288  CALL bedload2 neta = gbload! / (density! * discharge! ‘evaluate objective function  *  slope!)  END SUB FUNCTION shearforce ‘calculates the proportion of shear force on the banks shearforce!  =  1.766  *  (pbed! / pbank!  +  1.5)  A  -1.4026  END FUNCTION SUB stabcurve ‘stability curves are from Figure 5.3 thetadegrees!  =  theta! / 2 / 3.14159  *  360  iF phidegrees! <20 THEN ‘use phiprime = 15 degree stability curve from Figure 5.3 ‘from Taylor (1948) SELECT CASE thetadegrees! CASE 48.65 TO 90 stabnum! = -.141  thetadegrees!  +  17.46  CASE 29.36 TO 48.6499 stabnum! = -.533 * thetadegrees!  +  36.53  *  CASE 20.12 TO 29.3599 stabnum! = -3.07 * thetadegrees! CASE 15.001 TO 20.11999 stabnum! = -189.6 CASE IS  <=  111  +  *  thetadegrees!  15 stabnum!  =  1000000  END SELECT ELSE ‘use phiprime =25 degrees curve from Figure 5.3 After Taylor, (1948) SELECT CASE thetadegrees! CASE 58.62 TO 90 stabnum! = -.212 * thetadegrees!  +  CASE 39.11 TO 58.61999 stabnum? = -.8975 * thetadegrees!  24.86  +  65.04  CASE 25 TO 39. 10999  289  +  3844  stabnum  =  -6.97  *  thetadegrees!  +  302.5  CASE IS <=25 stabnum?  =  1000000  END SELECT END IF  END SUB  FUNCTION surfwidth Tunction to calculate the surface width surfwidth?  =  pbed? + COS(theta’)  *  pbank  END FUNCTION  FUNCTION velocity! ‘calculates the mean velocity rhbank! = bankshear! / (gamma! * slope!) rhbed! = bedshear! / (gamma! * slope!) Ibank! = (2.03 * LOG(12.2 * rhbank! / ksbank!) / LOG(10)) -2 ibed! = (2.03 * LOG(12.2 * rhbed! / ksbed!) / LOG(10)) A -2 ‘  ifactor!  =  velocity!  (ibed!  =  (8  *  *  pbed! / (pbed!  gravity!  *  +  hydrad!  pbank!) + fipJç! *  slope!)  A  •5  *  *  pbank! / (pbank!  ifactor!  END FUNCTION  290  +  pbed!)) A  APPENDIX C  FULL MODEL FORMULATION  The Full Model Formulation differs from the Bankfull:Variable Slope Model in that the fill flow duration data is used as input, the sediment transporting capacity of the channel is calculated over one year using the modified Parker (1990) surface-based sediment transport relation, 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 output data to a data file named rivmod.out. The input file contains data for one channel only and must be set-up as follows (for noncohesive bank sediment):  291  m  Q Q Q Q.  Gb k3d k$bk banktype 1.0 U 1 Q Probability of Being Equalled or Exceeded Probability of Q Being Equalled or Exceeded Probability of Q Being Equalled or Exceeded Probability of Q’ Being Equalled or Exceeded  D 1 D D D D  0 0 Proportion Finer than Proportion Finer than Proportion Finer than Proportion Finer than  n  Qbf  D  D D D’ D  1.0  D5Ob,  For noncohesive channel banks the banktype is equal to “n”, and for cohesive channels it is equal to “c”.  For cohesive banks the last line of the data file becomes: Ti  C’  crit  In rivmod.dat m is the number of flow intervals in the numerical approximation of a flow duration curve (See Figure 7.1); n is the number of intervals in the numerical approximation of a sediment gradation curve (See Figure 7.2);  Q  is the discharge at the lower bound of flow  interval 1 which is the lowest flow on record (or an approximate value),  292  Q,  is the discharge at  the upper bound of flow interval m and therefore the maximum flow on record (or an approximation); D is equal to D , and D is equal to D 0 . In both cases the superscript 1 or u 1 refer to the lower or upper bound of the interval, and the subscript indicates the interval number. The upper bound of one interval is equal to the lower bound of the next interval. The data for the input file is designed to be read directly from flow duration and sediment gradation curves. 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 in the List of Symbols.  The data values must be separated by at least one space although the number of spaces is not important. An example of an input file is given below: 85 2500000 11 7 3 1 22 0.2473 42 0.1064 55 0.0569 65 0.0365 75 0.0247 85 0,0172 90 0.01505 205 0.0095 115 0.007 125 0.0054 205 0 0.002 0 0.0064 0.1 0.01 0.2 0.018 0.4 0.032 0.6 0.06 0.8 0.091 0.9 0.15 1 0.07 40  0.4  0.4  ii  The output data will be output to rivmod. out in the following order:  293  W  H  17*  S  50 D  9  50 T*D  The output data from the sample input data file given above is: * W H D50 theta T*D50 S 37.7 1.29 1.16 0.00427 0.073 19.1 0.043  The source code for the Full Optimisation Model is presented below.  ‘OPTIMISATION MODEL BASED ON FULLY DEVELOPED MODEL DESCRIBED IN PhD THESIS BY RG MILLAR ‘UNIVERSITY OF BRiTISH COLUMBIA ‘SEPTEMBER 1994 ‘CALCULATES THE OPTIMUM (OR EQUILIBRIUM) CHANNEL GEOMETRY DECLARE SUB stabcurve  0  DECLARE SUB bedload2 0 DECLARE SUB bankstabilitynoncohesive 0 DECLARE SUB bankstabiitycohesive 0 DECLARE SUB parkerl99O 0 DECLARE SUB capG 0 DECLARE SUB vaiytheta 0 DECLARE SUB optimum 0 DECLARE SUB bedload 0 DECLARE FUNCTION bankshear! 0 DECLARE FUNCTION bedshear! 0 DECLARE FUNCTION shearforce! 0 DECLARE SUB continuity (discharge!) DECLARE FUNCTION velocity! 0 DECLARE FUNCTION hydrad! 0 DECLARE FUNCTION meandepth! 0 DECLARE FUNCTION area! 0 DECLARE FUNCTION surfwidth! 0 DECLARE FUNCTION depth! 0 program to calculate curves to demonstrate the optimum COMMON SHARED gravity!, gamma!, ss!, viscosity!, density!, meandischargel COMMON SHARED slope!, pbed!, pbank!, theta!, ddSO!, d50bank! COMMON SHARED qbfl, gbcalc!, gbload!, phiprime!, stability!, phidegrees COMMON 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 ‘ss!= specific gravity of sediment ‘density!=density of water ‘pbed!=bed perimeter  gamma! unit weight of water viscosity!=kinematic viscosity of water slope!channel slope pbank!bank perimeter  294  ‘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 of bed roughness ksbank!=measure of bank roughness ‘banktype$=type of bank sediment Wo=interger counter ‘gammat!=unit weight of cohesive bank sediment ‘stabnum!=stability number for cohesive sediment ‘cohesion!=soil cohesion taucrit!=critical shear stress ‘meandischarge!=mean annual discharge ‘phisgol=dimensionless bed shear stress (Eqn Eqn 4.31) ‘sigmaphi!=grainsize dispersion of armour (Eqn 4.37) ‘sigmaphitrial!=trial value of sigmaphil ‘dsg!=geometric mean grain diameter of pavement layer (Eqn 4.35) ‘capgamtna! = constant converts kg/sec to kg/year ‘capitalgl=dimensionless bedload function Eqn (4.32) ‘j%ith discharge j%jth sediment size ‘m%=numder of flow intervals n%=number of sediment size intervals  ‘SET VALUES OF INDEPENDENT CONSTANTS pi! = 3.14159 gravity! = 9.81 gamma! = 9810 viscosity! = .000001 density! = 1000 ss! =2.65 capganuna! = 365.25  *  24  *  3600  CLS PRINT PRINT PRINT  RIVERMOD AN OPTIMISATION MODEL FOR” PRINT” PRINT” PREDICTING THE ADJUSTMENT OF ALLUVIAL RIVERS” PRINT PRINT PRINT PRJNT” WR1TEN BY” ROBERT MILLAR” PRINT” PRINT PRINT” UNIVERSITY OF BRITISH COLUMBIA” PRINT PRINT PRINT PRINT HiT <ENTER> TO CONTINUE” PRINT” SLEEP INPUT dummy OPEN “nvmod.dat” FOR INPUT AS #1  295  OPEN “rivmod.out” FOR OUTPUT AS #2 iNPUT #1, m%, n%, qbfl, gbload!, ksbed!, ksbanld, banktype$ ‘input number of flow and sediment size intervals respectively ‘bankfufl discharge, mean annual sediment load bed and bank roughness heights ‘and bank type IF bankt,e$ = “c” OR banktype$ = “C” THEN banktype$ = “cohesive” ELSEIF banktype$ = “N” OR banktype$ = “n” THEN banktype$ = “noncohesive” ELSE BEEP CLS PRINT PRINT PRiNT PRiNT “ERROR iN DATA FILE” END END 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 variables i% =0 DOWHILEi%<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 flow i% = i% + 1 LOOP  j% =0 DO WHILEj% < n% INPUT #1, upperd(j%), propflner(j%) ‘input sediment data into array ‘upper bound of sediment fraction an dvolumetric proportion j%() corresponds to lower bound ie dO j%% corresponds to dlOO j%=j%+ 1 LOOP  CLS PRINT “DATA SUMMARY” PRINT PRINT  296  PRiNT” Discharge Data” PRINT”i Qi pi” ‘calculate geometric mean flow and probability for each interval i% = 1 qprob(O) =0 DO WHILE i% <= m% discharge(i%) = (upperq(i%) * upperq(i% - 1)) ‘ .5 qprob(i%) = probexceed(i% - 1) - probexceed(i%) PRINT USiNG “## ##L# #.###“; i%; discharge(i%); qprob(i%)  i% = i% + 1 LOOP PRINT PRINT HIT <ENTER> TO CONTINUE” PRINT” SLEEP INPUT dununy$ CLS PRINT “DATA SUMMARY” PRINT PRINT PRINT” Sediment Data” PRINT”j Di fl” ‘calculate geometric mean grain diameter for each grain interval ‘and volumetric proportion within each interval  j% =  1 f(O) = 0 DO WHTLEj% <= n% sediment(j%) = (upperd(j%) * upperd(j% 1)) A f(j%) = propfiner(j%) - propfiner(j% - 1) PRINT USING “#41 #.###41 #.###“; j%; sediment(j%); f(j%) j%=j%+ 1 LOOP PRINT -  ‘calculate mean annual discharge  i% = 1 totalq!=O DOWHILEi%<=m% totalq! = totalq! + discharge’(i%) i% = i% + 1 LOOP  *  ciprob(i%)  meandischarge! = totalq! ##*#.# m”3/sec”; meandischarge PRINT USING ‘Mean Annual Discharge PRINT USING “Mean Annual Sediment Load II111111111111 .41 kg/year”; gbload’ ‘input bank stability parameters IF banktype$ = “cohesive” OR banktype$ = “COHESIVE” THEN banktype$ = “cohesive” ELSEIF banktype$ = “NONCOHESIVE” OR banktype$ = “noncohesive” THEN banktype$ = “noncohesive” ELSE PRINT “Banktype unknoii: Error in Data File” BEEP CLOSE #1  297  CLOSE #2 END END IF PRINT PRINT “Bank Sediment Type:  IF banktype$ = “noncohesive” THEN ‘input bank stability parameters INPUT #1, d5Obank!, phidegrees! PRINT USING “D50 bank PRINT USING “Phi  #.### rn”; d50bank! ##.# degrees”; phidegrees!  ELSE ‘banktype$=”cohesive” INPUT #1, gzmmt!, phidegrees!, cohesion!, taucrit!  PRiNT USING “ganunat PRINT USING “Phi  ##.## kN/m”2”; gammat! / 1000 #W.# degrees”; phidegrees! #### J/jA”; cohesion! / 1000 ##.# N/m’2”; taucrit!  PRINT USING “cohesion PRiNT USING “Tau crit END IF  phiprime!  =  phidegrees!  *  2  *  pi! I 360  PRINT PRINT : INPUT “IS INPUT DATA OK (y/n) “; booleanS IF boolean$ = “n” OR boolean$ = “N” THEN  CLOSE #1 CLOSE #2 END END IF CLS PRINT PRINT  (upperd(0) + upperd(n%)) /2 ‘set trial dsg value at midpoint of sediment gradation sigmaphitrial! = 1 ‘set trial sigamphi value dsgtrial!  =  ‘COMMENCE OPTIMIZATION ROUTINE CALL optimum PRINT * PRINT” W H D50 theta T*D50” S PRINT USING “###.# #4.## ##.#4 #.IIihf liii #.### ##.# #.###“; surfwidth!; depth!; meandepth!; slope!; dd5O!; theta! * 360 / 2 / pi!; bedshear! / (gamma! * dd5O! * (! - 1)) Y* PRINT #2,” W H D50 theta T*D50” S  298  PRiNT #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 variables  CLOSE #1 CLOSE #2 END ,**********************************************************************************  FUNCTION area ‘function to calculate the cross-sectional area area!=.5*(pbed!+surfwidth!)*depth! END FUNCTION  FUNCTION bankshear ‘calculates the bank shear stress bankshear!  =  gamma!  *  depth!  *  slope!  *  shearforce!  *  ((surfwidth!  END FUNCTION  SUB bankstabilitycohesive ‘satisfies bank stability constraint for cohesive banks heightcond$ = “unknown” shearcond$ = “unknown” bankcond$ = “unknown” thetamax!=90*2*pi!/360 thetamin! =0 ‘initialise search and convergence criteria  Tirst test if vertical bank is stable wrt bank height theta! = thetamax! CALL continuity(qbf!) ‘satisfy continuity CALL stabcuive ‘catculate stability number criticalheight! = stabnum! * cohesion! / gmmatI ‘calculate the critical height stabilityl! = depth / criticaiheight! ‘calculates stability criteria wit bank height ‘if stabilityl! <= criticaiheight then bank is stable ‘with respect to bank height IF stabilityl! <= 1.001 THEN heightcondS = “stable”  299  +  pbed!)  *  S1N(theta!) / (4  *  depth!))  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! + thetamin!) /2 ‘calculate midpoint of range for ‘bisectrix convergence scheme  CALL continuity(qbfl) ‘satisfy continuity constraint CALL stabcurve ‘calculate stability number criticaiheight! = stabnum! * cohesion! / ganimat! stability 1! = depth / criticalheight! ‘calculates critical height and stabilty number ‘for the bank height constraint  IF stabilityl! > 1.001 THEN heightcond$ = “unstable” thetamax! = theta! ELSEIF stabiityl! <= .999 THEN heightcond$ = “understable” thetamin! = theta! ELSE heightcond$ = ‘just stable” ‘Bank Height = Critical Height EN])IF IF (thetainax! thetainin!) / theta! <.001 THEN heightcond$ = ‘just stable” stabilityl! = 1 ‘Secondary Convergence Criterion in case of ‘convergence problems due to numerical scheme END IF -  LOOP END IF  ‘bank height constraint now satisifled ‘now assess the bank shear constraint thetamin!=0 thetaniax! = theta! ‘thetaniax is reset to maximum bankangle which satisfied the ‘bank height constraint above stabilily2! = bankshear! / taucrit! ‘satbility criterion for bank shear constraint 300  IF stability2! < 1.001 THEN ‘bank shear constraint is satisified shearcond$ = “stable” bankcond$ = “stable”  ELSE ‘must reduce theta! DO UNTIL shearcond$  =  “just stable”  theta! =(thetaniax! +thetaniiu!)/2 ‘calculate mid point IF theta! <10 * 2 * 3.14159/360 THEN bank angles less than 10 degrees ‘nominally assumed to be unstable shearcond$ = “unstable” bankcond$ = “unstable” EXIT DO END IF CALL continuity(qbfi) ‘satisfy continuity = bankshear! / taucrit! ‘calculate stability criterion for bank shear constraint  stability2!  IF stability2! >= 1.001 THEN ‘bankshear > taucnt shearcond$ = “unstable” thetainax! =theta! ELSEIF stabiity2! <= .999 THEN shearcond$ “understable” thetaniin! = theta! ELSE shearcondS = “just stable” bankcond$ = “stable” END IF IF (thetamax! thetaniin!) / theta! <.001 THEN shearcond$ = “just stable” bankcond$ = “stable” stability2! =1 END iF -  LOOP END IF END SUB  301  ‘**********************************************************************************  SUB bankstabiitynoncohesive ‘calculates theta where banks just stable bankcond$ = “unknown” thetamax! = phiprime’ thetaniin! =0 ‘initialise search bounds and convergence criteria  DO UNTIL bankcond$ = ‘just stable” theta =(thetamax! +thetaniin!)/2 ‘determine midpoint of search range IF theta! <5*2*3.14159/360 THEN bankcond$ = “unstable” EXIT DO ‘if theta is less than 5 degrees the channel is assumed ‘to be unstable END IF CALL continuity(qbfl) ‘satisfies continuity constraint for qbf stability! = (bankshear! / (gamma! SIN(phiprime!) A 2) .5) ‘calculates stability criterion  *  (ss!  -  1)  *  d50bank!)) / (.048  IF stability! >= 1.001 THEN bankcond$ = “unstable” thetamax! =theta! ELSEEF stability! <= .999 THEN bankcond$ = “stable” thetamin! = theta! ELSE bankcond$ = “just stable” ‘pnmaiy convergence criterion END IF IF (thetamax! thetamin!) / theta! <.001 THEN bankcond$ = “just stable” stability! = 1 ‘secondaiy convergence criterion ENDIF -  LOOP  END SUB  SUB bedload ‘satisfies the bedload constraint using the Parker 1990  302  *  TAN(phiprime’)  *  (1 S1N(theta!) “2/ -  ‘surface-based sediment transport relation gbcalc! =0 minslope! =0 maxslope! = .05 ‘set bound of slope search  slopecond$ = “unknown” ‘initialise slope condition DO WHILE slopecond$  “satisfied”  slope! = (minslope! + maxslope!) /2 ‘calculate mid point of range for bisectrix convergence scheme IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE ‘banktype$ = “cohesive” CALL bankstabiitycohesive ‘solve bank stability constraint for cohesive banks END IF IF bankcond$ “unstable” THEN ‘bank stability is satisfied CALL parker 1990 ‘calculate sediment transporting capacity of trial channel IF gbcalc! <.999 * gbload! THEN ‘compares calculated to imposed sediment load slopecond$ = “too flat” minslope! = slope! ELSEIF gbcalc! > 1.001 * gbload! THEN slopecond$ = “too steep” maxslope! = slope! ELSE slopecond$ = “satisfied” ‘primary convergence criterion ‘sediment transporting capacity of the channel ‘equals the imposed load : bedload constraint satisfied EN]) IF IF (maxslope! minslope!) / slope! <.00001 THEN lFgbcalc! >.99*gbload! THEN slopecond$ = “satisfied” ‘secondary convergence criterion ‘when bounds of slope very small: convergence attained ‘occasionally a problem with convergence due to numerical ‘approximations ELSE slopecond$ = “not satisfied” EXIT DO ‘indicates that trial Pbed value is too narrow END IF -  303  END IF ELSE  maxslope! = slope! ‘trial slope too steep: reduce upper bound slopecond$ = “unstable” END IF LOOP  END SUB  SUB 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 difference minslope! = •9 * slope! maxslope! = 1.1 * slope! ‘set bound of slope search slopecond$ = “unknown” ‘initialise slope condition DO WHILE slopecond$ <> “satisfied” slope! = (minslope! + maxslope!) /2 ‘calculate mid point of range for bisectrix convergence scheme IF banktype$ = “noncohesive” THEN CALL bankstabiitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE banktype$ = “cohesive” CALL bankstabiitycohesive ‘solve bank stability constraint for cohesive banks END IF IF bankcond$ “unstable” THEN ‘if bank stability is satisfied CALL parker 1990 ‘calculate sediment transporting capacity of trial channel IFgbcalc! <.999*gbload! THEN ‘compares calculated to imposed sediment load slopecond$ = Htoo flat” minslope! = slope! ELSEIF gbcalc! > 1.001 * gbload! THEN slopecond$ = “too steep” maxslopel = slope! ELSE slopecondS = “satisfied” ‘sediment transporting capacity of the channel ‘equals the imposed load : bedload constraint satisfied END IF  304  IF (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 ‘approximations -  ELSE ‘if bank stability constraint not satisfied rnaxslope! = slope! ‘trial slope too steep: reduce upper bound slopecond$ = “unstable” END iF  LOOP END SUB ‘**********************************************************************************  FUNCTION bedshear ‘calculates the bed shear stress bedshear!  =  ganuna!  *  depth?  *  slope!  *  (1 shearforce!) -  *  ((surfwidth! / (2  END FUNCTION  SUB capG ‘calculate straining function SELECT CASE phisgo! ‘calculate omegao! based on piecewise linearisation of ‘function given in Parker (1990) (See figure 4.10 in thesis) CASE IS <= 1.033 omegao! = 1.011 CASE IS <= 5.44 omegao! = 1.027 * phisgo! A -.483 CASE IS > 5.44 omegao! = .453 END SELECT  SELECT CASE phisgo! ‘calculate sigmaphio based on piecewise linearisation of ‘function given in Parker (1990) (See figure 4.10 in thesis) CASE IS < .985 sigmaphio = .816 CASE IS <= 2.19 sigmaphio! = .395 * phisgo! + .426 CASE IS <=7 sigiuaphio’ = .044 * phisgo! + 1.194 CASE IS>7 signiaphio! = 1.501 END SELECT  305  *  pbed!)  +  .5))  straining!  =  1 + sigmaphil I sigmaphio! ‘Eqn4.38  *  (omegao!  -  1)  ‘calculate hiding function hiding!  =  (sediment(j%) / dsg!) A -.095 1 ‘Eqn 4,34 in thesis  modifiedphi! = phisgo! * hiding! * straining! ‘modifiedphi! is the product inside the square brackets ‘in Eqn 4.33  SELECT CASE modifiedphi! ‘calculate Capital G: dimensionless bedad function ‘Eqn 4.32 in thesis CASE IS < 1 capitaig! =modifiedphi! A 14.2 CASE IS <= 1.59 capitalg! = EXP(14.2 * (modifiedphi! 1) 9.28 * (modifiedphi! CASE IS> 1.59 capitaig! =5474*(1853/mjfipbj!)A45 END SELECT -  -  END SUB  SUB continuity (discharge!) ‘varies pbank! for trial values of Pbed!, slope!, and theta! to ‘satisfy the contiuity constraint pbankcond$ = “unknown” errorcalcl = 1000 minpbank! =0 maxpbank! = 20 * discharge! A 35 ‘initialise search and convergence criteria  DO UNTIL pbankcond$ pbank!  =  “OK”  (minpbank! + maxpbank!) /2 ‘calculate midpoint  =  errorcalc! = (area * velocity / discharge!) ‘calculate the normalised error IF errorcaic! > 1.001 THEN pbankcond$ = “too large” niaxpbank! = pbank! ELSEIF errorcalc! <.999 THEN pbankcond$ = “too small” minpbank! = pbank! 306  -  1)  A  2)  ELSE pbankcond$  =  “OK”  END IF  IF (maxpbank! minpbank!) / pbank! <.0001 AND pbankcond$ ‘resets maxpbánk! if too small maxpbank! =2* maxpbank! END IF LOOP -  END SUB  FUNCTION depth ‘ftmction to calculate flow depth of a trapezoidal channel depth!  =  .5  *  S1N(theta!)  *  phank!  END FUNCTION  FUNCTION hydrad! ‘function to calculate the hydraulic radius hydrad! =area! /(pbed! +pbank!) END FUNCTION  FUNCTION meandepth ‘function to calculate the mean depth meandepth!  =  area / surfwidth  END FUNCTION  SUB optimum ‘determines the optimal geometry optcond$ = “unknown” ‘initialises optimality test condition lowerpbed! =0 upperpbed! = 10 * qbf! “.5 ‘set bounds for Pbed based on Regime Eqns ‘typical optimal value of Pbed is 2 to 5 * discharge” .5 minpbed! = lowerpbed! maxpbed! = upperpbed! PRINT PRINT DO WHILE optcond$ <> “optimum” 307  = “too  small” THEN  pbed!  (minpbed! + maxpbed!) /2 ‘calculate mid point of search range  =  PRINT “Assessing Trial Bed Perimeter “; PRINT USING “#### ##“; pbed PRINT “m “; pbed! = pbed! * .975 ‘calculate backwards difference value of Pbed IF p  95 * upperpbed! THEN upperpbed! =2 * upperpbedl maxpbed’ = upperpbed! ‘reset upper bound if necessaly >  END IF CALL bedload ‘satisfies continuity, bank stability and bedload ‘constraints for the trail Pbed value IF slopecond$ = “satisfied” AND bankcond$ <> “unstable” THEN ‘test conditions indicate that stable channel geometty ‘has been determined and the bedload constraint ‘has been satisified netal!  gbload! I (capgamma! * density! * meandischarge! ‘evaluate nets for backward difference point  =  *  slope!)  pbed! =pbed! 1.975 * 1.025 ‘calculate pbed for forward duff  CALL 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 computations neta2!  pbed!  gbload! / (capgainma! * density! * meandischarge! ‘calculates nets for forward difference value  =  *  pbed! / 1.025 ‘reset pbed to midpoint value  =  dnetabydpbed! = (neta2! netal!) / (.05 * phed!) ‘calculate first derivative by finite difference ‘to assess optimum condition -  IF dnetabydpbed! <0 THEN ‘Trial Pbed is too wide maxpbed! =pbedl ‘ReduceupperboundofPbed optcond$ “too wide” ELSEIF dnetabydpbed! >0 THEN ‘Trial Pbed is too narrow minpbed! = pbed! ‘Increase lower bound of Pbed optcond$ = “too narrow” ELSE 308  slope!)  optcond$ END IF  =  “optimum”  ‘Optimum Achieved  IF (maxpbed nunpbedl) / pbed’ <.001 THEN optcond$ = “optimum” ‘Convergence attained. Second optimality criterion -  ELSE trial Pbed too small for stable geometry: Increase minimum Pbed pbed! =pbed’ 1.975 ‘reset from backward difference to correct value minpbed! = pbed! ‘set lower limit at current trial value ‘(as the optimum value must be greater) optcond$ = “too narrow” END IF PRINT optcond$ LOOP  ‘Evaluate geometry at exact optimal Pbed ‘(not at forward or backward difference value) IF banktype$ = “noncohesive” THEN CALL bankstabilitynoncohesive ‘solve bank stability constraint for noncohesive banks ELSE ‘banktype$ = “cohesive” CALL bankstabilitycohesive ‘solve bank stability constraint for cohesive banks END IF CALL 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 computations  END SUB ‘*********************************************************************************,  SUB parkerl99O DIM 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 = 1 DOWH[LEi%<=m% discharge! = discharge(i%) IF discharge! > qbfl THEN discharge!  =  qbfl 309  ‘sets all flows in excess of qbf equal to qbfl ‘in order to calculate sediment transport rate ‘for overbank flows CALL 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% + 1  LOOP dsgcond$ = “unknown” sigmaphicondS = “unknown” ‘initialise convergence criteria for dsg! and sigmaphi! DO WHILE dsgcond$ “converged” OR sigmaphicondS dsg!=dsgtrial! sigmaphi! = sigmaphitrial!  “converged”  OMEGA! =0 j% = 1 DOWIIILEj%<=n% = 1 OMEGAJj%) =0 DOWHILEi%<=m% ‘calculate for sediment(j%) over range of flows phisgo! = bedsheari(i%) / (gamma! * (ss! 1) * dsg!) / .0386 CALL capG OMEGAIJ(i%, j%) = qprob(i%) * capitaig! * bedsheari(i%) A 1.5 ‘OMEGAIJ(m%) is used in the summation over i% in the numerator ‘of Eqn 4.52 -  OMEGAJ(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% + 1  LOOP OMEGA! = OMEGA! + f(j%) I OMEGAJ(j%) ‘OMEGA is the denomenator of Eqn 4.52  j%=j%+ 1 LOOP  ‘calculate Fj and Dsg and sigmaphi! j% = 1 lndsg! =0 310  sigmaphi2! =0 ff1 =0 DOWHILEj%<=n% ff(j%) = f(j%) / OMEGAJ(j%) / OMEGA! Indsg! = lndsg! + ff(j%) * LOG(sediment(j%))  sigmaphi2!  =  sigiuaphi2!  +  (LOG(sediment(j%) / dsgl) /LOG(2)) “2  j%=j%+ 1 LOOP dsg! = EXP(lndsg!) sigmaphi! = sigmaphi2! “.5 ‘test for convergence for dsg! IF (dsg! / dsgtrial!) < .99 THEN dsgcond$ = “too small” ELSEIF (dsg! I dsgtrial!)> 1.01 THEN dsgcond$ = “too large” ELSE dsgcond$ = “converged” END IF ‘test for convergence for siginaphi! IF (sigmaphi! / sigmaphitrial!) < .99 THEN sigmaphicond$ = “too small” ELSEIF (signiaphi! / sigmaphitrial!)> 1.01 THEN sigmaphicondS = “too large” ELSE sigmaphicond$ = “converged” END IF sigmaphitrial! = sigmaphi dsgtnal! dsg! LOOP ‘loop until convergence for dsg! and signaphi!  ‘calculate the median grain diameter ff(0) = 0 sediment(0) =0 j% =0 totif! =0 DO WHILE totifi <.5 1 totifi = totfP + ff(j%) LOOP  j%=j%+  deldd50! = (totifi .5) * (upperd(j%) upperd(j% 1)) / ff(j%) ddSO! = upperd(j%) deldd50! ‘calculate D50 by interpolation -  -  -  -  qbcalc’  =  .00218 I ((ss! 1) * gravity! * densityV 1.5 * OMEGA!) ‘calculate the mean unit volumetric sediment transport rate ‘in units of m”2/sec -  311  *  ff(j%)  gbcalc!  =  qbcalc? * pbed! * density! * ss! * capganuna! ‘cacluate the total load in mass units ‘transported over one year.  END SUB  FUNCTION shearforce ‘calculates the proportion of shear force on the banks shearforce!  =  1.766  *  (pbed! /pbank!  +  1.5)  A  -1.4026  END 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  *  360  IF phidegrees! <20 THEN ‘use phiprime = 15 degree stability curve from Figure 5.3 ‘from Taylor (1948) SELECT CASE thetadegrees! CASE 48.65 TO 90 stabnum! = -.141  thetadegrees?  +  17.46  CASE 29.36 TO 48.6499 stabnum! = -.533 * thetadegrees?  +  36.53  *  CASE 20.12 TO 29.3599 stabnum! = -3.07 * thetadegrees!  +  111  CASE 15.001 TO 20.11999 stabnum! = -189.6 * thetadegrees!  +  CASE IS <=15 stabnum! = 1000000 ‘bank of infinite height is stable END SELECT ELSE ‘use phiprinie =25 degrees curve from Figure 5.3 ‘After Taylor, (1948) SELECT CASE thetadegrees! CASE 58.62 TO 90 stabnum! = -.2 12 * thetadegrees!  +  24.86 312  3844  CASE 39.11 TO 58.61999 stabnum! = -.8975 * thetadegrees!  +  65.04  CASE 25 TO 39.10999 stabnum! = -6.97 * thetadegrees!  +  302.5  CASE IS <=25 stabnuml = 1000000 bank of infinite height is stable END SELECT END IF  END SUB  FUNCTION surfwidth ‘function to calculate the surface width surfwidth! = pbed’ END FUNCTION  +  COS(theta!)  *  pbank!  FUNCTION velocity! ‘calculates the mean velocity rhbank! = bankshear! / (gamma! * slope?) rhbed? = bedshear! / (gamma! * slope!) Ibank! = (2.03 * LOG(12.2 * rhbank! / ksbank!) / LOG(10)) “-2 Ibed! = (2.03 * LOG(12.2 * rhbed! / ksbed!) / LOG(10)) A -2 ffactor2!  =  (Ibed!  *  pbed! / (pbed!  velocity! = (8 * gravity! END FUNCTION  *  hydrad!  +  pbank!) + thank!  *  slope!) A 5  *  *  pbank! / (pbank!  fl’actor2!  313  +  pbed!))  A  APPENDIX D  DATA FROM HEY AND THORNE (1986) AND ANDREWS (1984)  Table D-1 contains the data from Hey and Thorne (1986). The “Observed” columns contain the actual published data: Qbfi W,  Y, D 50 and Vegetation Type. The values of k 3 and  Gbf are  calculated from the observed hydraulic geometry using Eqn (3.2) and the Einstein-Brown Equation (Eqn 6.2) respectively. The value Of  is set equal to D , 5 0 and 5 d 0 is set equal to  /3. 5 D 0  Table D-2 contains the data from Andrews (1984). The “Observed” columns contain the actual published data:  QbJ;  W, Y,  d 5 0 and Vegetation Type. The values of k and Gbf are  calculated from the observed hydraulic geometry using Eqn (3.2) and the Einstein-Brown Equation (Eqn 6.2) respectively. The value of  is set equal to 1)5o. Channel Number  9074800 from Andrews (1984) was excluded from the analysis. (See Section 6.3.3).  314  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  Channel Number  (mi/s) 104 154 124 17 63 68 70 148 76.1 172 304 45 50 46 53 53 69 48 24 24 22 14 7.1 10 28  Qbf  k 3 (m) 0.16 0.11 0.26 0.60 0.20 0.07 1.34 0.04 0.24 0.07 0.11 0.54 0.85 0.07 0.19 0,63 0.10 0.12 0.21 0.66 0.53 0.14 0.46 0.21 0.34  Gbf (kg/s) 101.55 94.43 40.96 4.97 293.86 13.83 450.75 24.29 430.24 22.75 35.12 27.59 1341.0 19.96 30.17 100.32 61.08 170.87 21.24 41.89 30.65 50.46 13.02 90.59 33.00 25.2 42.7 32.7 12.8 20.0 21.6 27.5 29.4 15.8 28.7 48.5 18.2 18.4 20.1 25.2 24.4 29.7 13.0 17.7 21.1 10.2 10.6 13.7 5.5 17.0  W (m)  Y (m) 1.52 1.40 1.70 0.91 1.27 1.36 1.15 1.94 1.47 2.19 2.49 1.41 1.14 1.05 1.02 1.16 1.02 1.07 0.80 0.92 1.42 0.75 0.48 0.59 0.99  Observed  0.0035 0.0031 0.0025 0.0045 0.0042 0.0022 0.0127 0.0014 0.0064 0.0016 0.0013 0.0030 0.0133 0.0027 0.0039 0.0051 0.0036 0.0082 0.0040 0.0034 0.0022 0.0039 0.0061 0.0214 0.0036  S  0 D 5 (m) 0.057 0.061 0.082 0.109 0.024 0.061 0.176 0.041 0.046 0,070 0.066 0.067 0.060 0.043 0.075 0.071 0.045 0.065 0.048 0.034 0.019 0.014 0.048 0.109 0.043  50 d (m) 0.019 0.020 0.027 0.036 0.008 0.020 0.059 0.014 0.015 0.023 0.022 0.022 0.020 0.014 0.025 0.024 0.015 0.022 0.016 0.011 0.006 0.005 0.016 0.036 0.014 D5obO (m) 0.057 0.061 0.082 0.109 0.024 0.061 0.176 0.041 0.046 0.070 0,066 0.067 0.060 0.043 0.075 0,071 0.045 0.065 0.048 0.034 0.019 0.014 0.048 0.109 0.043  W (m) 41.1 45.4 33.9 11.6 71.7 21.4 37.6 38 59.7 31 47.2 25.6 74.4 24 23.8 34.2 35.9 32.2 20.3 31.6 37 37.4 13.7 13.4 25.4  III II III II III IV I IV IV IV III III IV II II 11 II IV II 1 IV IV I IV II  ,  Type  Veg  Table D-1. Data from Hey and Thorne (1986).  =40° I (m) 1.15 1.35 1.69 0.99 0.62 1.38 0.99 1.68 0.69 2.12 2.58 1.18 0,52 0.96 1.07 0.97 0,92 0.64 0.75 0.75 0.7 0.37 0.49 0.36 08 (°) 55.5 41.9 41.3 35.9 72.5 39.7 48.7 49.5 75.8 46.3 37.5 55 73 46 38 50.1 45.2 67 44.1 51.2 90 72.5 39.7 73.9 52,8  Fixed S Est. Mean ç W I (m) (m) 1.48 27.1 1.56 36 2.04 25.1 1.08 10 44.6 0.81 1.7 15.9 1.13 29.2 23.5 2.31 1.07 28.2 2.58 23.5 3.08 36.1 1.44 18.4 0.79 33.2 1.09 19.5 19.8 1.2 26.8 1.12 1.06 28.1 0.96 16.2 0.85 16.2 0.78 29.1 1.05 18.2 0.57 17.6 0.51 12.8 0.52 7.2 0.92 20  Calculated  45.1 46.2 34.2 11.4 89.8 22.5 39.4 39.2 72.6 31.8 47.5 26.4 91.3 24.6 23.9 36 37.4 37 20.9 33.4 40.6 46.3 13.7 14.4 267  W (m)  Variable S çb, —40° S I (m) 0.0041 1.04 0.0032 1.32 0.0026 1.66 0.0045 1 0.0064 0.48 0.0023 1.32 0.0137 0.95 0.0015 1.6 0.0092 0.56 2,04 0.0017 0.0013 2.57 0.0032 1.14 0.0184 0.44 0.0029 0.92 0.0039 1.06 0.0056 0.92 0.0039 0.87 0.0105 0.55 0.0042 0.72 0.0038 0.71 0.0026 0.64 0.0058 0.29 0.0062 0.48 0.0249 0.33 0.0040 0.75  26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50  Number  Channel  k, (m)  0.32 0.32 0.21 0.17 0.45 0.30 0.17 0.21 0.32 0.44 1.51 0.38 1.12 1.47 0.10 0.36 0.29 0.50 0.59 0.38 0.13 1.28 0.87 0.42 0.34  27 34 100 100 7.5 7.5 170 7.1 9.7 35 11.2 53 237 424 61 61 45 75 120 348 60.4 36.5 74.7 95 50  Qbf  (mi/s)  Gbf  40.89 8.38 61.97 31.40 6.92 0.07 116.42 1.91 39.83 671.82 22.71 302.58 79.13 316.91 23.29 90.48 107.% 87.73 348.10 438.07 3.70 126.72 666.58 29.13 98.13  (kg/s) 15.0 14.4 25.9 23.0 15.6 12.6 31.8 9.6 12.3 12.5 14.1 15.2 57.8 76.5 22.9 27.7 18.6 20.4 41.7 46.6 32 2 28.0 19.3 32.2 17.5  W (m) 0.95 1.61 1.52 1.70 0.48 0.56 1.82 0.75 0.48 0.93 0.81 1.18 2.60 3.10 1.20 1.29 1.12 1.58 1.25 2.51 1.31 0.99 1.53 1.40 1.20  Y (m)  Observed  0.0048 0.0013 0.0035 0.0027 0.0052 0.0034 0.0033 0.0015 0.0108 0.0152 0.0055 0.0092 0.0014 0.0016 0.0027 0.0026 0.0046 0.0042 0.0066 0.0030 0.0011 0.0057 0.0073 0.0038 0.0055  S 0.055 0.027 0.081 0.084 0.055 0.086 0.079 0.017 0.066 0.068 0.074 0.077 0.056 0.050 0.053 0.028 0.042 0.081 0.091 0.066 0.036 0.064 0.058 0.130 0.070  D 5 0 (m) 0.018 0.009 0.027 0.028 0.018 0.029 0.026 0.006 0.022 0.023 0.025 0.026 0.019 0.017 0.018 0.009 0.014 0.027 0.030 0.022 0.012 0.021 0.019 0.043 0.023  0 d., (m) D 5 0 (m) 0.055 0.027 0.081 0.084 0.055 0.086 0.079 0.017 0.066 0.068 0.074 0.077 0.056 0.050 0.053 0.028 0.042 0.081 0.091 0.066 0.036 0.064 0.058 0.130 0.070  Table D- I continued.  Veg II III IV IV 1 1 IV II 1 IV 1 IV I I IV II II IV I IV I I IV II III  Type  W (m) 23.1 26.3 32.2 27.1 12.1 7.1 43.1 13.7 15.4 46.7 18 39.3 65.5 117.9 26 52.1 38.8 33.5 53.5 87.2 23.5 38 69.4 24.5 30.4  =40° 1’ (m) 0.75 1.15 1.35 1.57 0.56 0.8 1.53 0.62 0.43 0.46 0.73 0.7 2.45 2.46 1.13 0.91 0.74 1.21 1.1 1.76 1.61 0.86 0.78 1.66 0.88 (°) 54.5 70.3 47.3 49.5 30.9 21.1 50.1 55 46.3 73.6 49.8 67.7 44.1 51.7 44.1 57.7 61 58.3 46.8 58.3 30.4 48.4 73.6 31.5 58.3  Fixed S Est. Mean W (m) 18.4 18.4 19.7 19.5 11.6 6.9 25.1 11.3 14.3 21.5 17.2 19.1 61.6 109 16.1 40.2 29.9 19.3 49.5 43.8 22.3 35.1 31.8 22.3 20.1  (m) 0.86 1.44 1.85 1.96 0.57 0.82 2.16 0.7 0.45 0.7 0.75 1.06 2.54 2.57 1.54 1.05 0.86 1.7 1.14 2.66 1.66 0.89 1.18 1.77 1.13  ç , 6 Y  Calculated  W (m) 24.3 26.7 33.2 28.9 11.7 7 45.4 14.1 16 56.6 18.1 45.3 66.1 126.8 26.4 58.7 44 35.4 56.4 97.6 22.8 39.7 82.6 23.7 34.2  I (m) 0.71 1.12 1.29 1.49 0.58 0.86 1.44 0.6 0.41 0.38 0.72 0.61 2.41 2.29 1.1 0.8 0.65 1.14 1.04 1.56 1.69 0.82 0.66 1.74 079  Variable S  0.0054 0.0014 0.0038 0.0029 0.0048 0.0027 0.0037 0.0016 0.01 15 0.0212 0.0057 0.0117 0.0015 0.0018 0.0029 0.0032 0.0057 0.0047 0.0072 0.0037 0.0010 0.0061 0.0098 0.0035 0.0064  S  51 52 53 54 55 56 57 58 59 60 61 62  Channel Number  (ms/s) 90.5 126.8 358.3 91.3 153.3 192.3 17.6 196 19 20 3.9 28  Qbf  k 3 (m) 0.57 1.25 0.13 0.10 0.09 0.67 0.12 0.37 0.24 0.05 0.15 0.05  Gk, (kg/s) 32.64 227.40 70.31 26.77 9.55 86.21 18.18 334.75 24.41 23.22 0.67 2.28  W (m) 24.6 31.3 77.1 31.2 33.4 45.2 9.1 41.0 11.6 17.5 6.5 12.3  Y (m) 1.73 2.03 2.01 1.41 1.98 2.08 0.73 1.70 1.05 0.62 0.59 1.14  Observed  0.0033 0.0036 0.0016 0.0019 0.0014 0.0025 0.0084 0.0047 0.0024 0.0035 0.0019 0.0018  S 0.123 0.071 0.060 0.041 0.069 0.092 0.114 0.084 0.018 0.020 0.023 0.049  D 5 0 (m)  d 5 0 (m) 0.041 0.024 0.020 0.014 0.023 0.031 0.038 0.028 0.006 0.007 0.008 0.016 0 D 5 (m) 0.123 0.071 0.060 0.041 0.069 0.092 0.114 0.084 0.018 0.020 0.023 0.049  Table D- 1 continued.  Veg Type IV III I II III 111 IV III 111 I 11 III 24.8 59.1 61 34.9 28.6 47 10.7 60.5 31 27.4 7.4 13  (m)  çó. W  =40° Y (m) 1.76 1.46 2.32 1.33 2.23 2.06 0.68 1.37 0.61 0.48 0.56 1.13 39.7 59.1 33.4 43.5 34.2 40.8 52.3 51.2 70.3 52.8 46.3 430  (9  ,..  Est.  Fixed S Mean Y W (m) (m) 2.05 19.8 1.88 37.4 2.41 57.4 1.52 28.1 2.64 22.3 2.53 33.6 0.82 8.1 1.79 38.5 0.79 19.4 0.5 25.5 0.62 6.4 1.33 10.3  Calculated  25.1 63.9 59.7 35.6 28.4 47.2 11.2 65.1 35.5 30.1 7.6 13.1  W (m)  (m) 1.72 1.34 2.39 1.29 2.26 2.03 0.65 1.27 0.53 0.43 0.54 1.11  ‘  0.0034 0.0042 0.0015 0.0020 0.0014 0.0026 0.0089 0.0054 0.0030 0.0041 0.0020 0.0019  Variable S =40°  00  6611900 6614800 6620000 7083000 7091000 9013500 9018000 9022000 9034800 9035900 9036000 9078100 9078200 9081600 9112500 9115500 9124500 9242500 9244410 9249500 9251000 9253000 9257000  Channel Number  Qbf  6.71 0.7 85.2 7.08 9.77 12.2 2.21 2.69 1.87 8.36 22.6 3.17 2.52 49 37.5 7.08 42 101 167 46.7 255 72.2 114  (mIs)  0.03 0.17 0,02 0.83 0.45 0.40 0.62 0.24 0.13 0.18 0.16 0.34 0.50 0.46 0.83 0.29 0.34 0.64 0.21 1.39 0.12 0.69 0.43  3 k (m)  Gbf (kg!s) 2.90 7.85 13.01 129.82 222.01 12.12 10.61 30.52 2.16 123.25 19.61 10.10 10.88 169.69 66.22 7.06 151.66 114.83 46.77 16.82 51.53 2184.75 42.66  W (m) 10.4 2.3 47.2 9.1 11.5 11.9 7.3 7.0 5.2 9.2 17.5 6.3 5.6 34.1 26.0 11.6 24.9 36.6 53.3 24.4 83.8 30.5 36.6 0.48 0.20 0.97 0.52 0.46 0.73 0.34 0.29 0.31 0.43 0.73 0.39 0.41 0.84 0.91 0.52 0.88 1.45 1.63 1.62 1.85 1.13 1.65  (in)  Y  Observed  0.0023 0.0260 0.0014 0.0170 0.0190 0.0046 0.0110 0.0140 0.0061 0.0150 0.0044 0.0100 0.0092 0.0058 0.0067 0.0046 0.0058 0.0037 0.0018 0.0020 0.0009 0.0071 0.0024  S D.so (m) 0.023 0.049 0.024 0.079 0.085 0.061 0.052 0.034 0.024 0.073 0.043 0.061 0.037 0.098 0.091 0.046 0.079 0.064 0.058 0.045 0.034 0.122 0.070  0 d, (m) 0.005 0.019 0.008 0.023 0.019 0.019 0.019 0.011 0.011 0.013 0.017 0.019 0.015 0.015 0.036 0.014 0.014 0.024 0.017 0.022 0.007 0.007 0.025  D 5 0 (m) 0.023 0.049 0.024 0.079 0.085 0.061 0.052 0.034 0.024 0.073 0.043 0.061 0.037 0.098 0.091 0.046 0.079 0.064 0.058 0.045 0.034 0.122 0.070 Thin Thick Thin Thick Thick Thin Thin Thick Thick Thick Thick Thick Thick Thin Thin Thin Thin Thin Thin Thin Thin Thin Thin  Veg Type  Table D-2. Data from Andrews (1984).  çS. =40° W Y (m) (m) 0.5 9.7 0.13 5.1 1.16 36.1 0.37 18.7 18.6 0.36 0.68 14 0.3 9.5 0.21 13 0.26 7.3 13.4 0.35 0.66 21.1 0.34 8.3 0.29 11.4 1.01 25.2 27.6 0.89 0.53 11.4 0.88 25.4 1.25 48.5 1.85 43.4 1.41 32.6 58.3 2.3 32.4 1.1 1.65 37.5 37.5 67.6 33.1 60.5 52.8 46.3 47.9 56.6 52.8 50.6 45.7 50.6 63.8 32.0 41.9 39.7 40.8 47.9 33.9 48.4 30.4 41.9 40.8  (0)  .  Fixed S Est. Mean W (m) 9.6 3 36 10.5 10.5 13.9 9.5 7.2 4.8 8 13 5.5 6.7 25.3 27.5 11.4 25.3 47.9 43 32.5 57.9 32.3 37.5  Y (m) 0.5 0.18 1.16 0.49 0.49 0.68 0.3 0.29 0.34 0.47 0.89 0.44 0.38 1.01 0.9 0.53 0.88 1.26 1.87 1.41 2.3 1.11 1.65  Calculated  9.6 5.4 34.7 20.2 20 14 9.7 14.4 7.8 14.4 22.1 8.4 12 24.9 28.1 11.5 25.7 52 42.9 35.4 56.3 33.4 38.2  (in)  W  —  Y (m) 0.5 0.13 1.22 0.35 0.34 0.67 0.29 0.19 0.25 0.33 0.63 0.33 0.27 1.03 0.88 0.52 0.86 1.18 1.89 1.3 2.41 1.07 1.61  =400  S 0.0024 0.0293 0.0013 0.0190 0.0213 0.0048 0.0115 0.0165 0.0067 0.0168 0.0048 0.0106 0.0102 0.0055 0.0069 0.0047 0.0060 0.0040 0.0017 0.0023 0.0008 0.0076 0.0025  Variable S  APPENDIX E  DATA FROM CHARLTON ETAL. (1978)  Table E-1 contains the data from Chariton eta!. (1978) for channels with cohesive banks which  was analysed in Chapter 6. The actual published data are: Q W, 7, S, D. , 50  q,  and the  Vegetation Type. The values for k are calculated from the observed hydraulic geometry using Eqn (3.2), and c’ is calculated from qu using Eqn (6.4) with the assumption of ç ’ 4  =  25° for all  channels. The value of d 50 is set equal to /3, 50 and the drained and saturated values of the unit D soil weight are assumed to be 20.0 and 22.45 3 kN/m respectively for all channels.  The modelled output data is presented in Table 6.4.  319  t%) C  Reach Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14  AforLwyd Aiwen 1-5 Alwen5-9 Arrow Ceirog 1-3 Ceirog 4-7 Exe IrfonC Otter Teign Trent Usk Wye Wrye  River (m’/s) 64 10 10.7 29.5 66 66 67 81 14.2 66 2.7 157 550 38  W (m) 17.4 14 9.8 13.7 17.6 19 31 28.7 16.7 19 5.2 39.3 59.4 19.5 Y (m) 1.78 0.73 0.73 1.34 1.79 1.36 1.77 1.63 0.69 2.47 0.65 2.64 4.19 1.67 0.0044 0.0064 0.0130 0.0045 0.0048 0.0105 0.0018 0.0014 0.0032 0.0014 0.0023 0.0009 0.0007 0.0020  S (m) 1.09 1.42 1.25 1.16 1.21 1.07 1.35 0.19 0.29 1.41 0.56 0.60 0.28 1.54  D, (in) 0.075 0.106 0.113 0.041 0.071 0.082 0.043 0.055 0.057 0.051 0.033 0.072 0.028 0.040 kN/m 2 39.5 25.6 23.2 51.4 23.3 70.9 26.1 30.1 25.8 15.7 36.7 21.3 16.0 39.9 kN/m 2 12.6 8.2 7.4 16.4 7.4 22.6 8.3 9.6 8.2 5.0 11.7 6.8 5.1 12.7  el a!. (1978).  d,o (m) 0.025 0.035 0.038 0.014 0.024 0.027 0.014 0.018 0.019 0.017 0.011 0.024 0.009 0.013  Table E-1. Data from Chariton  (°) 25 25 25 25 25 25 25 25 25 25 25 25 25 25  kN/m 3 22.45 22.45 22.45 22.45 22.45 22.45 22.45 22.45 22.45 22.45 22.45 22.45 22.45 22.45  kN/m 3 20 20 20 20 20 20 20 20 20 20 20 20 20 20  Veg Type T G T T T T G T G T G G G G  

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