1. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 1 MORPHODYNAMICS OF SAND-BED RIVERS ENDING IN DELTAS Wax Lake Delta, Louisiana Delta of Eau Claire River at Lake Altoona, Wisconsin Delta from Iron Ore Mine into Lake Wabush, Labrador

2. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 2 DELTA SHAPE Most deltas show a 2D spread in the lateral direction (fan-delta). The channel(s) migrate and avulse to deposit over the entire surface of the fan-delta as they prograde into standing water. Mangoky River, Malagasy But some deltas forming in canyons can be approximated as 1D. The delta of the Colorado River at Lake Mead was confined within a canyon until recently. Colorado River at Lake Mead, USA 1D progradation 2D progradation

3. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 3 AN EXAMPLE OF A 1D DELTA Hoover Dam was closed in 1936. Backwater from the dam created Lake Mead. Initially backwater extended well into the Grand Canyon. For much of the history of Lake Mead, the delta at the upstream end has been so confined by the canyon that is has propagated downstream as a 1D delta. As is seen in the image, the delta is now spreading laterally into Lake Mead, forming a 2D fan-delta. View of the Colorado River at the upstream end of Lake Mead. Image from NASA https://zulu.ssc.nasa.gov/mrsid/mrsid.pl

4. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 4 HISTORY OF SEDIMENTATION IN LAKE MEAD, 1936 - 1948 Image based on an original from Smith et al. (1960)

5. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 5 WHERE DOES THE SAND GO? WHERE DOES THE MUD GO? Image based on an original from Smith et al. (1960)

6. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 6 STRUCTURE OF A DELTA: TOPSET, FORESET AND BOTTOMSET A typical delta deposit can be divided into a topset, foreset and bottomset. The topset is coarse-grained (sand or sand and gravel), and is emplaced by fluvial deposition. The foreset is also coarse-grained, and is emplaced by avalanching. The bottomset is fine-grained (mud, e.g. silt and clay) and is emplaced by either plunging turbidity currents are rain from sediment-laden surface plumes. standing water

7. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 7 (Kostic and Parker, 2003a,b)

8. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 8 SIMPLIFICATION: TOPSET AND FORESET ONLY Here the problem is simplified by considering a topset and foreset only. That is, the effect of the mud is ignored. It is not difficult to include mud: see Kostic and Parker (2003a,b). standing water

9. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 9 DERIVATION OF THE 1D EXNER EQUATION OF SEDIMENT CONSERVATION The channel has a constant width. Let x = streamwise distance, t = time, q t = the volume sediment transport rate per unit width and p = bed porosity (fraction of bed volume that is pores rather than sediment). The mass sediment transport rate per unit width is then  s q t, where  s is the material density of sediment. Mass conservation within a control volume with length  x and a unit width (Exner, 1920, 1925) requires that:  /  t (sediment mass in bed of control volume) = mass sediment inflow rate – mass sediment outflow rate or control volume

10. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 10 BUT HOW ARE FLOODS ACCOUNTED FOR IN THE EXNER EQUATION OF SEDIMENT CONTINUITY? Fly River, PNG in flood

11. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 11 SIMPLE ADAPTATION TO ACCOUNT FOR FLOODING Rivers move the great majority of their sediment, and are morphodynamically active during floods. Paola et al. (1992) represent this in terms of a flood intermittency I f. They characterize floods in terms of bankfull flow, which carries a volume bed material transport rate per unit width q t for whatever fraction of time is necessary is necessary to carry the mean annual load of the river. That is, where G ma is the mean annual bed material load of the river, B bf is bankfull width and T a is the time of one year, I f is adjusted so that where  = water density. The river is assumed to be morphodynamically inactive at other times. Wright and Parker (2005a,b) offer a specific methodology to estimate I f. The Exner equation is thus modified to transport rate at bankfull flow fraction of time flow is in flood

12. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 12 KEY FEATURES OF THE MORPHODYNAMICS OF THE SELF-EVOLUTION OF 1D SAND-BED DELTAS UNDER THE INFLUENCE WITH BACKWATER Water discharge per unit width q w is conserved, and is given by the relation and shear stress is related to flow velocity using a Chezy (recall C f -1/2 = C) or Manning-Strickler formulation; In low-slope sand-bed streams boundary shear stress cannot be computed from from the depth-slope product, but instead must be obtained from the full backwater equation; or thus

13. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 13 DOWNSTREAM BOUNDARY CONDITION FOR THE BACKWATER FORMULATION Base level is specified in terms of a prescribed downstream water surface elevation  d (t) =  (L, t) +H(L, t) rather than downstream bed elevation  (L, t). The base level of the Athabasca River, Canada is controlled by the water surface elevation of Lake Athabasca. Delta of the Athabasca River at Lake Athabasca, Canada. Image from NASA https://zulu.ssc.nasa.gov/mrsid/mrsid.pl

14. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 14 THE MORPHODYNAMIC PROBLEM The formulation below assumes a single characteristic bed grain size D, a constant Chezy resistance coefficient Cz = C f -1/2 and the Engelund-Hansen (1967) total bed material load formulation as an example. Exner equation Bed material load equation Backwater equation Specified initial bed profile Specified upstream bed material feed rate Downstream boundary condition, or base level set in terms of specified water surface elevation.

15. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 15 NUMERICAL SOLUTION TO THE BACKWATER FORMULATION OF MORPHODYNAMICS The case of subcritical flow is considered here. At any given time t the bed profile  (x, t) is known. Solve the backwater equation upstream from x = L over this bed subject to the boundary condition Evaluate the Shields number and the bed material transport rate from the relations Find the new bed at time t +  t Repeat using the bed at t +  t

16. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 16 Honey, could you scratch my back, it itches in a place I can’t reach. Sure, sweetie, but could you cut my toenails for me afterward? I can’t reach ‘em very well either. IN MORPHODYNAMICS THE FLOW AND THE BED TALK TO AND INTERACT WITH EACH OTHER

17. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 17 THE PROBLEM OF IMPULSIVELY RAISED WATER SURFACE ELEVATION (BASE LEVEL) AT t = 0 M1 backwater curve Note: the M1 backwater curve was introduced in the lecture on hydraulics and sediment transport q ta

18. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 18 RESPONSE TO IMPULSIVELY RAISED WATER SURFACE ELEVATION: A PROGRADING DELTA THAT FILLS THE SPACE CREATED BY BACKWATER See Hotchkiss and Parker (1991) for more details.

19. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 19 FLOW OVER ANTECEDENT BED Before the water surface is raised, the is assume to be at normal, mobile-bed equilibrium with antecedent bed slope S a and water dischage per unit width during floods q w. It is useful to compute the characteristics of the normal flow that would prevail with the specified flow over the specified bed. Let H na and q tna denote the flow depth and total volume bed material load per unit width that prevail at antecedent normal mobile-bed equilibrium with flood water discharge per unit width q w and bed slope S a. From Slides 19 and 21 of the lecture on hydraulics and sediment transport, these are given as

20. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 20 CHANGES IMPOSED AT t = 0 The reach has length L and the bed has antecedent bed slope S a. It is assumed for simplicity that the antecedent bed elevation at the downsteam end of the reach (x = L) is  da = 0, so that antecedent water surface elevation  da is given as At time t = 0 the water surface elevation (base level) at the downstream end  d is impulsively raised to a value higher than  da and maintained indefinitely at the new level. In addition, the sediment feed rate from the upstream end is changed from the antecedent value q ta to a new feed value q tf.

21. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 21 ULTIMATE EQUILIBRIUM Eventually the backwater zone (e.g. reservoir zone behind a dam) fills with sediment and the river established a new normal, mobile-bed equilibrium in consonance with the flood, water discharge per unit width q w, the sediment supply rate q tf (which becomes equal to the transport rate q tu at ultimate equilibrium) and the water surface elevation  d. The bed slope S u and flow depth H nu at this ultimate normal equilibrium can be determined by solving the two equations below for them. A morphodynamic numerical model can then be used to describe the evolution from the antecedent equilibrium to the ultimate equilibrium

22. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 22 NUMERICAL MODEL: INITIAL AND BOUNDARY CONDITIONS The channel is assumed to have uniform grain size D and some constant ambient slope S a (before changing conditions at t = 0) which is in equilibrium with an ambient transport rate q ta. The reach of interest has length L. The antecedent bed profile (which serves as the initial condition for the calculation) is then where here  da can be set equal to zero. The boundary condition at the upstream end is the changed feed rate q tf for t > 0, i.e. where q tf (t) is a specified function (but here taken as a constant). The downstream boundary condition, however, differs from that used in the normal flow calculation, and takes the form where  d (t) is in general a specified function, but is here taken to be a constant. Note that downstream bed elevation  (L,t) is not specified, and is free to vary during morphodynamic evolution.

23. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 23 NUMERICAL MODEL: DISCRETIZATION AND BACKWATER CURVE The reach L is discretized into M intervals of length  x bounded by M+1 nodes. In addition, sediment is fed in at a “ghost” node where bed elevation is not tracked. The backwater calculation over a given bed proceeds as in the lecture on hydraulics and sediment transport: where i proceeds downward from M to 1.

24. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 24 Note that the difference scheme used to compute  q t,i /  x is a central difference scheme only for a u = 0.5. With a backwater formulation takes an advectional-diffusional form and a value of a u greater than 0.5 (upwinding) is necessary for numerical stability. The difference  q t,1 is computed using the sediment feed rate at the ghost node: NUMERICAL MODEL: SEDIMENT TRANSPORT AND EXNER

25. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 25 INTRODUCTION TO RTe-bookAgDegBWChezy.xls The worksheet RTe-bookAgDegBWChezy.xls provides both a grapical user interface and code in Visual Basic for Applications (VBA). A tutorial on VBA is provided in the workbook Rte-bookIntroVBA; it introduces the concept of modules. The code for the morphodynamic model is contained in Module 1 of Rte- bookAgDegBWChezy.xls. It can be seen by clicking “Tools”, “Macros”, “Visual Basic Editor” from Excel, and then double-clicking “Module1” in the VBA Project Window at the upper left of the Screen. The Security Level (“Tools”, “Macro”, “Security”) must be set to no higher than “medium” in order to run the code. Most of the input is specified in worksheet “Calculator”. The first set of input includes: water discharge per unit width q w at flood, flood intermittency I f, grain size D, reach length L, Chezy resistance coefficient Cz, antecedent bed slope S a and volume total bed material feed rate per unit width during floods q tf. The specified numbers are then used to compute the normal flow depth H na at antecedent conditions, the final ultimate equilibrium bed slope S u and the final ultimate equilibrium normal flow depth H nu. The user then specifies a downstream water surface elevation  d. This value should be > the larger of either H na or H nu to cause delta formation.

26. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 26 INTRODUCTION TO RTe-bookAgDegBWChezy.xls contd. The following input parameters are then specified on worksheet “Calculator” by the user: reach length L, time step  t, the number of time steps until data is generated for output (by printing it onto another worksheet in the workbook) Ntoprint, the number of times data is generated Nprint, number of spatial intervals M and upwinding parameter a u. The total duration of the calculation is thus equal to  t x Ntoprint x Nprint, and the spatial step length  x = equal to L/M. The parameter R is specified in worksheet “AuxiliaryParameter”. Once all the input parameters are specified, the code is executed by clicking the button “Do the Calculation” in worksheet “Calculator”. The numerical output is printed onto worksheet “ResultsofCalc”. The output consists of the position x, bed elevation , water surface elevation  and flow depth H at every node for time t = 0 and Nprint subsequent times. The bed elevations and final water surface elevations are plotted on worksheet “PlottheData”.

27. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 27 INTRODUCTION TO RTe-bookAgDegBWChezy.xls contd. In worksheet “Calculator” the flow discharge q w (m 2 /s) and bed material feed rate at flood flow q tf (m 2 /s) are specified per unit channel width. In worksheet “MeanAnnualFeedRate” the user can specify a channel width B f at flood flow (e.g. bankfull width). The flood discharge Q f = q f B f in m 3 /s and the mean annual bed material feed rate G ma are then computed directly on the worksheet. The input for all the cases (Cases A ~ G) illustrated subsequently in this presentation is given in worksheet “WorkedCases”. As noted in Slide 25, the code itself can be viewed by clicking “Tools”, “Macros” and “Visual Basic Editor”, and then double-clicking Module 1 in the VBA Project Window in the upper left of the screen. Each unit of the code is termed a “Sub” or a “Function” in VBA. Three of these units are illustrated in Slides 29, 30 and 31.

28. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 28 GRAPHICAL USER INTERFACE The graphical user interface in worksheet “Calculator” is shown below.

29. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 29 Sub Do_Fluvial_Backwater() Dim Hpred As Double: Dim fr2p As Double: Dim fr2 As Double: Dim fnp As Double: Dim fn As Double: Dim Cf As Double Dim i As Integer H(M + 1) = xio - eta(M + 1) For i = 1 To M fr2p = qw ^ 2 / g / H(M + 2 - i) ^ 3 Cf = (1 / alr ^ 2) * (H(M + 2 - i) / kc) ^ (-1 / 3) fnp = (eta(M + 1 - i) - eta(M + 2 - i) - Cf * fr2p * dx) / (1 - fr2p) Hpred = H(M + 2 - i) - fnp fr2 = qw ^ 2 / g / Hpred ^ 3 fn = (eta(M + 1 - i) - eta(M + 2 - i) - Cf * fr2 * dx) / (1 - fr2) H(M + 1 - i) = H(M + 2 - i) - 0.5 * (fnp + fn) Next i For i = 1 To M xi(i) = eta(i) + H(i) Next i End Sub BACKWATER CALCULATION: Sub Do_Fluvial_Backwater

30. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 30 Sub Find_Shields_Stress_and_Load() Dim i As Integer Dim taux As Double: Dim qstarx As Double: Dim Cfx As Double For i = 1 To M + 1 taux = Cfx * (qw / H(i)) ^ 2 / (Rr * g * D) qstarx = alt/cFS *taux ^ 2.5 qt(i) = ((Rr * g * D) ^ 0.5) * D * qstarx Next i End Sub LOAD CALCULATION: Sub Find_Shields_Stress_and_Load

31. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 31 Sub Find_New_eta() Dim i As Integer Dim qtback As Double: Dim qtit As Double: Dim qtfrnt As Double: Dim qtdif As Double For i = 1 To M If i = 1 Then qtback = qtf Else qtback = qt(i - 1) End If qtit = qt(i) qtfrnt = qt(i + 1) qtdif = au * (qtback - qtit) + (1 - au) * (qtit - qtfrnt) eta(i) = eta(i) + dt / (1 - lamp) / dx * qtdif * Inter Next i qtdif = qt(M) - qt(M + 1) eta(M + 1) = eta(M + 1) + dt / (1 - lamp) / dx * qtdif * Inter time = time + dt End Sub IMPLEMENTATION OF EXNER: Sub Find_New_eta

32. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 32 CASE A This is a case for which a) base level  d is unaltered from the antecedent value, and b) the bed material feed rate q tf is equal to the antecedent normal equilibrium value q tna. Condition a) is ensured by setting  d equal to the antecedent normal equilibrium depth H na, and condition b) is ensured by setting the bed material feed rate q tf so that the ultimate equilibrium bed slope S u is equal to the antecedent bed slope S a. The result is the intended one: nothing happens over the 600-year duration of the calculation.

33. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 33 CASE B This case has the same input as Case A, except that the downstream water surface elevation is raised from 8.89 m to 20 m. The duration of the calculation is 6 years. A delta starts to form at the upstream end! delta!

34. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 34 CASE C Same conditions as Case B, but the time duration is 30 years.

35. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 35 CASE D Same conditions as Case B, but the time duration is 60 years.

36. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 36 CASE E Same conditions as Case B, but the time duration is 120 years.

37. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 37 CASE F Same conditions as Case B, but the time duration is 600 years. The “dam” is filled and overtopped, and ultimate normal mobile-bed equilibrium is achieved.

38. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 38 CASE G Same conditions as Case D, but the upstream feed rate q tf is tripled compared to the antecedent normal value q tna, so ensuring that S u is greater than S a This slope is steeper than this slope Note that the delta front has prograded out much farther than Case D because the sediment feed rate is three times higher.

39. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 39 GENERALIZATIONS The basic formulation can be generalized to include a self-formed channel with varying width, channel sinuosity, a channel-floodplain complex in which mud as well as sand can deposit, a foreset of specified slope and a 2D geometry that yields a fan shape to the delta. These generalizations are implemented for the Wax Lake Delta shown below in the paper “Large scale river morphodynamics: application to the Mississippi Delta” (Parker et al,. 2006) included on the CD for this course as file “WaxLake.pdf”.

40. Contribution from the National Center for Earth-surface Dynamics for the Short Course Environmental Fluid Mechanics: Theory, Experiments and Applications Karlsruhe, Germany, June 12-23, 2006 40 REFERENCES Exner, F. M., 1920, Zur Physik der Dunen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 129 (in German). Exner, F. M., 1925, Uber die Wechselwirkung zwischen Wasser und Geschiebe in Flussen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 134 (in German). Hotchkiss, R. H. and Parker, G., 1991, Shock fitting of aggradational profiles due to backwater, Journal of Hydraulic Engineering, 117(9): 1129 ‑ 1144. Kostic, S. and Parker, G., 2003a, Progradational sand-mud deltas in lakes and reservoirs. Part 1. Theory and numerical modeling, Journal of Hydraulic Research, 41(2), pp. 127-140. Kostic, S. and Parker, G.. 2003b, Progradational sand-mud deltas in lakes and reservoirs. Part 2. Experiment and numerical simulation, Journal of Hydraulic Research, 41(2), pp. 141-152. Paola, C., Heller, P. L. & Angevine, C. L., 1992, The large-scale dynamics of grain-size variation in alluvial basins. I: Theory, Basin Research, 4, 73-90. Parker, G., Sequeiros, O. and River Morphodynamics Class of Spring 2006, 2006, Large scale river morphodynamics: application to the Mississippi Delta, Proceedings, River Flow 2006 Conference, Lisbon, Portugal, September 6 – 8, 2006, Balkema. Smith, W. O., Vetter, C.P. and Cummings, G. B., 1960, Comprehensive survey of Lake Mead, 1948-1949: Professional Paper 295, U.S. Geological Survey, 254 p. Wright, S. and Parker, G., 2005a, Modeling downstream fining in sand-bed rivers. I: Formulation, Journal of Hydraulic Research, 43(6), 612-619. Wright, S. and Parker, G., 2005b, Modeling downstream fining in sand-bed rivers. II: Application, Journal of Hydraulic Research, 43(6), 620-630.