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CHAPTER 21 RESPONSE OF A SAND-BED RIVER TO A DREDGE SLOT

CHAPTER 21 RESPONSE OF A SAND-BED RIVER TO A DREDGE SLOT. Grey Cloud Island. Lock and Dam no. 2. Mississippi River near Grey Cloud Island south of St. Paul, Minnesota Image from NASA https://zulu.ssc.nasa.gov/mrsid/mrsid.pl. FILLING OF A DREDGE SLOT IN A SAND-BED RIVER.

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CHAPTER 21 RESPONSE OF A SAND-BED RIVER TO A DREDGE SLOT

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  1. CHAPTER 21 RESPONSE OF A SAND-BED RIVER TO A DREDGE SLOT Grey Cloud Island Lock and Dam no. 2 Mississippi River near Grey Cloud Island south of St. Paul, Minnesota Image from NASA https://zulu.ssc.nasa.gov/mrsid/mrsid.pl

  2. FILLING OF A DREDGE SLOT IN A SAND-BED RIVER The Mississippi River must be dredged in order to maintain a depth sufficient for navigation. In addition, gravel and sand are mined for industrial purposes on Grey Cloud Island adjacent to the river. Now suppose that a) gravel and sand mining is extended to a dredge slot between the island and the main navigation channel and b) the main channel subsequently avulses (jumps) into the dredge slot. How would the river evolve subsequently? dredge slot navigation channel

  3. FILLING OF A DREDGE SLOT IN A SAND-BED RIVER contd. before avulsion just after avulsion some time after avulsion Note how a) a delta builds into the dredge slot from upstream and b) degradation propagates downstream of the slot.

  4. MORPHODYNAMICS OF DREDGE SLOT EVOLUTION In the modeling performed here, the following form of the Exner equation of sediment continuity is used: where  = bed elevation, qt = total volume bed material load transport rate per unit width, qb = volume bedload transport rate per unit width and qs = volume bed material suspended load transport rate per unit width. Here qs is computed based on the assumption of quasi-steady flow. An alternative formulation from Chapter 4 is, however, where is the near-bed concentration of suspended sediment, E is the dimensionless rate of entrainment of suspended sediment from the bed and vs denotes the fall velocity of the sediment. That is, vs denotes the volume rate of deposition of suspended sediment per unit area per unit time on the bed, and vsE denotes the corresponding volume rate of entrainment of sediment into suspension from the bed per unit area per unit time.

  5. ALTERNATIVE ENTRAINMENT FORMULATION FOR EXNER If the alternative formulation for conservation of bed sediment is used, then a) the quasi-steady form of the equation of conservation of suspended sediment, i.e. must be solved simultaneously, and in addition the above relations must be closed by relating to C. For example, writing = roC, the following quasi-steady evalution is obtained from the material of Chapter 10: This alternative formulation is not used here. The reasons for this are explained toward the end of the chapter.

  6. BEDLOAD TRANSPORT AND ENTRAINMENT RELATIONS An appropriate relation for bedload transport in sand-bed streams is that of Ashida and Michiue (1972) introduced in Chapter 7. Where bs denotes the boundary shear stress due to skin friction and the relation takes the form An appropriate relation for the entrainment of sand into suspension is that of Wright and Parker (2004) introduced in Chapter 10: Note that in the disequilibrium formulation considered here, bed slope S has been replaced with friction slope Sf.

  7. RELATION FOR SUSPENDED SEDIMENT TRANSPORT RATE The method of Chapter 10, which is strictly for equilibrium flows, is hereby extended to gradually varied flows. A backwater calculation generates the depth H, the friction slope Sf and the depth due to skin friction Hs everywhere. Once these parameters are known, the parameters u* = (gHSf)1/2, Cz = U/u*, kc = 11H/[exp(Cz)], u*s = (gHsSf)1/2 and E can be computed everywhere. The volume transport rate per unit width of suspended sediment is thus computed as where In the case of the Wright-Parker formulation, b = 0.05.

  8. SUMMARY OF MORPHODYNAMIC FORMULATION In order to finish the formulation, a resistance relation that includes the effect of bedforms is required. This relation must be adapted to mildly disequilibrium flows and implemented in a backwater formulation.

  9. CALCULATION OF GRADUALLY VARIED FLOW IN SAND-BED RIVERS INCLUDING THE EFFECT OF BEDFORMS The backwater equation presented in Chapter 5 is where H denotes depth, x denotes downstream distance, S is bed slope and Sf is friction slope. In addition the Froude number Fr = qw/(g1/2H3/2) where qw is the water discharge per unit width and g is the acceleration of gravity. The friction slope can be defined as follows: In a sand-bed river, the boundary shear stress b and depth H can be divided into components due to skin friction bs and Hs and form drag due to dunes bf and Hf so where Cfs and Cff denote resistance coefficients due to skin friction and form drag.

  10. GRADUALLY VARIED FLOW IN SAND-BED RIVERS WITH BEDFORMS contd. As shown in Chapter 9, the formulation for Wright and Parker (2004) for gradually varied flow over a bed covered with dunes reduces to the form for skin friction and the form for form drag. Reducing the above equation using the friction slope Sf rather than bed slope S in order to be able to capture quasi-steady flow, If H, qw, Ds50, Ds90 and R are known, Hs and Sf can be calculated iteratively from the above equations. Note that a stratification correction in the first equation above (specified in Wright and Parker, 2004) has been set equal to unity for simplicity in the present calculation.

  11. LIMITS TO THE WRIGHT-PARKER FORMULATION The ratio s of bed shear stress due to skin friction to total bed shear stress can be defined as Now by definition s must satisfy the condition s≤ 1 (skin friction must not exceed total friction). The following equation for s is obtained from the Wright-Parker relation: For any given value of Froude number Fr, it is found that a minimum Shields number min* exists, below which s > 1. In order to include values * < min*, the relation must be amended to:

  12. PLOT OF s VERSUS * FOR THE CASE Fr = 0.2

  13. ITERATIVE COMPUTATION OF min* AS A FUNCTION OF Fr The equation for min* takes the form or This equation cannot be solved explicitly. It can, however, be solved implicitly using, for example the Newton-Raphson technique. Let p = 1, 2, 3… be an index, and let min,p* be an estimate of the root of the above equation. A better estimate is min,p+1*, where The calculation proceeds until the relative error  between min,p* and min,p+1* drops below some specified small tolerance  << 1, i.e.

  14. IMPLEMENTATION OF ITERATIVE COMPUTATION OF min* AS A FUNCTION OF Fr A sample calculation for the case Fr = 0.2 is given below.

  15. PLOT OF min* VERSUS Fr

  16. Sub Find_tausmin(xFr, xtausmin) (Dim statements deleted) xtausmin = 0.4 Found_tausmin = False Bombed_tausmin = False ittau = 0 Do ittau = ittau + 1 Ft = xtausmin - 0.05 - 0.7 * (xtausmin ^ (4 / 5)) * (xFr ^ (14 / 25)) Ftp = 1 - 0.7 * (4 / 5) * (xtausmin ^ (-1 / 5)) * (xFr ^ (14 / 25)) xtausminnew = xtausmin - Ft / Ftp er = Abs(2 * (xtausminnew - xtausmin) / (xtausminnew + xtausmin)) If er < ep Then Found_tausmin = True Else If ittau > 200 Then Bombed_tausmin = True Else xtausmin = xtausminnew End If End If Loop Until Found_tausmin Or Bombed_tausmin If Found_tausmin Then xtausmin = xtausminnew Else Worksheets("ResultsofCalc").Cells(1, 5).Value = "Calculation of tausmin failed to converge" End If End Sub CODE FOR COMPUTING min*

  17. CALCULATION OF Hs AND Sf FROM KNOWN DEPTH H In order to implement a backwater calculation that includes the effects of bedforms, it is necessary to compute Hs and Sf at every point for which depth H is given. The governing equations are: Now writing  = Hs/H, the top equation can be solved for Sf to yield

  18. CALCULATION OF Hs AND Sf FROM KNOWN DEPTH H contd. The middle equation of the previous slide can be reduced with the definition  =Hs/H and the last equation of the previous slide to yield where This relation reduces to

  19. NEWTON-RAPHSON SCHEME FOR s A Newton-Raphson iterative solution is implemented for s. Thus if p is an index and s,p is the pth guess for the solution of the above equation, a better guess is given by where the prime denotes a derivative with respect to s. Computing the derivative, The solution is initiated with some guess s,1. The calculation is continued until the relative error  is under some acceptable limit  (e.g.  = 0.001). where

  20. SAMPLE IMPLEMENTATION OF NEWTON-RAPHSON SCHEME The following parameters are given at a point: H = 3 m, qw = 5 m2/s, Ds50 = 0.5 mm, Ds90 = 1 mm, R = 1.65. Thus ks = 3Ds90 = 3 mm and An iterative computation of min* yields the value 0.113. Implementing the iterative scheme for s with the first guess s = 0.99, the following result is obtained: Note that the relative error  is below 0.1% by the fourth iteration (p = 5).

  21. NOTE OF CAUTION CONCERNING THE NEWTON-RAPHSON SCHEME • There always seems to be a first guess of s for which the Newton-Raphson scheme converges. When nom* is only slightly greater than min* (in which case s is only slightly less than 1), however, the right initial guess is sometimes hard to find. For example, the scheme may bounce back and forth between two values of s without converging, or may yield at some point a negative value of s, in which case Sf cannot be computed from the first equation at the bottom of Slide 17. • The following technique was adopted to overcome these difficulties in the programs presented in this chapter. • The initial guess for s is set equal to 0.9 • Whenever the iterative scheme yields a negative value of s, s is reset to 1.02 and the iterative calculation recommenced. • Whenever the calculation does not converge, it is assumed that s is so close to 1 that it can be set equal to 1. • These issues can be completely avoided by using the bisection method rather than the Newton-Raphson method for computing s. The bisection method, • however, is rather slow to converge.

  22. APPLICATION TO BACKWATER CALCULATIONS The backwater equation takes the form Let the water discharge qw, the grain sizes Ds50 and Ds90 and the submerged specific gravity R be specified. In addition the upstream and downstream bed elevations i-1 and i are known, along with the downstream depth Hi. The downstream values Hs,i and Sf,i are computed in accordance with the procedures of the previous three slides. A first guess of Hi-1 is given as Hi-1,pred, where or thus:

  23. APPLICATION TO BACKWATER CALCULATIONS contd. Once Hi-1,pred is known, the associated parameters Hs,i-1,pred and Sf,i-1,pred can be computed using the Newton-Raphson formulation outlined in previous slides. Having obtained these values, a predictor-corrector scheme is used to evaluate Hi-1: or thus: Once Hi-1 is known the Newton-Raphson scheme can be used again to compute Hs,i-1 and Sf,i-1.

  24. CALCULATION OF NORMAL DEPTH FROM GIVEN VALUES OF qw, D, R and S The calculations for the morphodynamic response to a dredge slot begin with a computation of the normal flow conditions prevailing in the absence of the dredge slot. It is assumed that the water discharge per unit width qw, bed slope S, sediment grain size D and sediment submerged specific gravity R are given parameters. The parameters H and Hs associated with normal flow are to be computed. The governing equations are the same as the first two of Slide 17, except that Sf = S at normal flow.

  25. CALCULATION OF NORMAL DEPTH contd. Again introducing the notation s = Hs/H, the first equation of the previous slide reduces to: The second equation of the previous slide then reduces with the above equation to

  26. CALCULATION OF NORMAL DEPTH contd. The second equation of the previous slide can also be rewritten as The corresponding Newton-Raphson scheme is where the term FN’ is given on the following slide. The initial guess for H can be based on the depth for normal flow that would prevail in the absence of bedforms (form drag only):

  27. CALCULATION OF NORMAL DEPTH contd. The scheme thus becomes For a given value Hp, the parameter min* is computed using the iterative method of Slide 13.

  28. CALCULATION OF NORMAL DEPTH AND BACKWATER CURVE: INTRODUCTION TO RTe-bookBackwaterWrightParker.xls All three iterative schemes (i.e. for min*; Sf and Hs; and normal depth Hn). are implemented in this workbook. The user specifies a flow discharge Qw, a channel width B, a median grain size D50, a grain size D90 such that 90% of the bed material is finer, a sediment submerged specific gravity R and a (constant) bed slope S. Clicking the button “Click to compute normal depth” allows for computation of the normal depth Hn. Downstream bed elevation is set equal to 0, so that at normal conditions the downstream water surface elevation d = Hn. The user may then specify a value of d that differs from Hn (as long as the corresponding downstream Froude number is less than unity), and compute the resulting backwater curve by clicking the button “Click to compute backwater curve”. The program generates a plot of bed and water surface elevations  and  versus streamwise distance, as well as a plot of depth H and depth due to skin friction Hs versus streamwise distance.

  29. SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: INPUT

  30. SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: OUTPUT

  31. SAMPLE CALCULATION WITH RTe-bookBackwaterWrightParker.xls: OUTPUT contd.

  32. CALCULATION OF DREDGE SLOT EVOLUTION: INTRODUCTION TO RTe-bookDredgeSlotBW.xls The Excel workbook RTe-bookDredgeSlotBW.xls implements the formulation given in the previous slides for the case of filling of a dredge slot. The code used in RTe-bookBackwaterWrightParker.xls is also used in RTe-bookDredgeSlotBW.xls. The code first computes the equilibrium normal flow values of depth H, depth due to skin friction Hs and volume bed load and bed material suspended load transport rates per unit width qb and qs for given values of flood water discharge Qw, flood intermittency If, channel width B, bed sediment sizes D50 and D90 (both assumed constant), sediment submerged specific gravity R and (constant) bed slope S. A dredge slot is then excavated at time t = 0. The hole has depth Hslot, width B and length (rd - ru)L, where L is reach length, ruL is the upstream end of the dredge slot. and rdL is the downstream end of the dredge slot. Once the slot is excavated, it is allowed to fill without further excavation. Specification of the bed porosity p, the number of spatial intervals M, the time step t, the number of steps to printout Mtoprint, the number of printout after the one corresponding to the initial bed Mprint and the upwinding coefficient au completes the input.

  33. SAMPLE CALCULATION OF DREDGE SLOT EVOLUTION: INPUT This calculation for a very short time illustrates the state of the bed just after excavation of the dredge slot.

  34. SAMPLE CALCULATION OF DREDGE SLOT EVOLUTION: RESULT The result indicates the backwater set up by the dredge slot.

  35. THE DREDGE SLOT 3 YEARS LATER Changing Mtoprint from 1 to 50 allows for a calculation 3 years into the future. The bed degrades both upstream and downstream of the slot as it fills.

  36. THE DREDGE SLOT 12 YEARS LATER 12 years later (Mtoprint from 1 to 200 the slot is nearly filled. Degradation upstream of the slot is 0.3 ~ 0.4 m, and downstream of the slot it is on the order of meters.

  37. THE DREDGE SLOT 24 YEARS LATER The slot is filled and the degradation it caused is healing.

  38. THE DREDGE SLOT 48 YEARS LATER The slot and its effects have been obliterated; normal equilibrium has been restored.

  39. A SPECIAL CALCULATION: DREDGE SLOT PLUS BACKWATER Consider the case of a very long reach:

  40. DREDGE SLOT PLUS BACKWATER: MODIFICATION TO CODE Some minor modifications to the code allow the specification of a depth of 20 m at the downstream end, insuring substantial backwater in addition to the dredge slot: Sub Set_Initial_Bed_and_Time() 'sets initial bed, including dredge slot Dim i As Integer: Dim iup As Integer: Dim idn As Integer For i = 1 To M + 1 x(i) = dx * (i - 1) eta(i) = S * L - S * dx * (i - 1) Next i time = 0 'xi(M + 1) = xid xid = 20 'debug: use this statement and the statement below to specify 'backwater at downstream end xi(M + 1) = xid 'debug

  41. DREDGE SLOT PLUS BACKWATER: 240 YEARS LATER Filling in the dredge slot retards filling in the backwater zone downstream (which might be due to a dam).

  42. DREDGE SLOT PLUS BACKWATER: 960 YEARS LATER The dredge slot is filled and the backwater zone downstream is filling.

  43. TRAPPING OF WASH LOAD A sufficiently deep dredge slot can capture wash load (e.g. material finer than 62.5 m) as well as bed material load. As long as the dredge slot is sufficiently deep to prevent re-entrainment of wash load, the rate at which wash load fills the slot can be computed by means of a simple settling model. Let Cwi = concentration of wash load in the ith grain size range, and vswi = characteristic settling velocity for that range. Wash load has a nearly constant concentration profile in the vertical, so that ro  1. Neglecting re-entrainment, the equation of conservation of suspended wash load becomes where Cuwi denotes the value of Cwi at the upstream end of the slot and Lsu denotes the streamwise position of the upstream end of the slot. Including wash load, Exner becomes As the slot fills, however, wash load is resuspended and carried out of the hole well before bed material load.

  44. WHY WAS THE ENTRAINMENT FORMULATION OF EXNER NOT USED? That is, why was not Exner implemented in terms of the relations for bed material load as given in Slide 5? The reason has to do with the relatively short relaxation distance for suspended sand. To see this, consider the following case: Qw = 300 m3/s, If = 1, B = 60 m, D50 = 0.3 mm, D90 = 0.8 mm, R = 1.65 and S = 0.0002. The associated fall velocity for the sediment is 0.0390 m/s (at 20 C). From worksheet “ResultsofCalc” of workbook RTe-bookDredgeSlotBW.xls, it is found that the depth-averaged volume concentration C is 5.97x10-5. The same calculation yields a value of E of 0.00705 m/s (must add a line to the code to have this printed out), and thus the quasi-equilibrium value of ro = E/C of 11.82.

  45. RELAXATION DISTANCE FOR SUSPENDED SEDIMENT PROFILE Now suppose that at time t = 0 this normal flow prevails everywhere, except that at x = 0 the sediment is free of suspended sediment. The flow is free to pick up sediment downstream of x = 0. As described in the pickup problem of Chapter 10 (and simplified here with a depth-averaged formulation), the equation to be solved for the spatial development of the profile of suspended sediment is The solution to this equation is

  46. RELAXATION DISTANCE FOR SUSPENDED SEDIMENT PROFILE contd. Further setting qw = Qw/B = 5 m2/s, ro = 11.82, vs = 0.0390 m/s and E = 0.00705, the following evaluation is obtained for C(x); Note that C is close to its equilibrium value of 5.97x10-5 by the time x = 30 m.

  47. RELAXATION DISTANCE FOR SUSPENDED SEDIMENT PROFILE contd. It is seen from the solution that the characteristic relaxation distance Lsr for adjustment of the suspended sediment profile is In the present case, Lsr is found to be 10.8 m. Whenever Ls is shorter than the spatial step length x used in the calculation, it is appropriate to assume that the suspended sediment profile is everywhere nearly adapted to the flow conditions, allowing the use of the formulation in place of the more complicated entrainment formulation. This is true for the cases considered in this chapter.

  48. MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION An accurate morphodynamic formulation satisfies mass conservation. That is, the total inflow of sediment mass into a reach must equal the total storage of sediment mass within the reach minus the total outflow of sediment mass from the reach. Consider a reach of length L. During floods, the mass inflow rate of bed material load is B(R+1)qt(0, t) and the mass outflow rate is B(R+1)qt(L, t) where qt denotes the volume bed material load per unit width. The Exner equation of sediment conservation is Integrating this equation from x = 0 to x = L yields the result

  49. MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION contd. In the present formulation is discretized to the form where I = i,new - I and In the above relation qtf is the sediment feed rate at x = 0 and au is an upwinding coefficient.

  50. MASS CONSERVATION OF A MORPHODYNAMIC FORMULATION contd. Summing the discretized relation from i = 1 to i = M+1 and rearranging yields the result Thus as long as the volume inflow rate of bed material load per unit width is interpreted as auqtf + (1 - au)qt,1 the method is mass-conserving: the volume storage of sediment in the reach per unit width in one time step = the volume input of sediment per unit width – the volume output of sediment per unit width, both over one time step. The method is specifically mass-conserving in terms of qtf if pure upwinding (au = 1) is employed. (This method is numerically less accurate than partial upwinding, however). Mass conservation is tested numerically on worksheet “ResultsMassBalance” of workbook RTe-booKDredgeSlotBW.xls.

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