ONE-DIMENSIONAL (1-D) FLOW AND SEDIMENT TRANSPORT NUMERICAL MODELS

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1 ONE-DIMENSIONAL (1-D) FLOW AND SEDIMENT TRANSPORT NUMERICAL MODELS Kamal EL KADI ABDERREZZAK EDF-R&D, Laboratoire National d Hydraulique et Environnement (LNHE) September 2009 UNL, Santa Fe, Argentina

2 OUTLINE I. Formulation of 1-D flow and sediment transport models - General principles - Flow hydrodynamic equations - Sediment transport equations - Lateral allocation of bed changes II. 1-D bank failure models III. 1-D models for dam-break waves over movable beds IV. Decoupled, coupled and semi-coupled techniques V. Data requirements of 1-D model VI. Examples of 1-D morphodynamic models 2

3 GENERAL PRINCIPLES 1-D models simulate the flow and sediment transport in the streamwise direction of a channel without solving the details over the cross-section 1-D models require the least amount of field data, and the numerical schemes used for solving the water and sediment governing equations are more stable and offer order of magnitude gains in computational time over 2- and 3-D models The selection of a 1-D model for each specific problem is usually contingent on the knowledge about the system, the available measurements, and the specific obectives of the study Applications: Long-term sediment transport problems in rivers, reservoirs, estuaries River restoration works, hydropower generation, flood control and disaster alleviation, water supply, environment enhancement Dam-break waves over movable beds 3

4 CLEAR-WATER EQUATIONS Basic assumptions «Concentration of sediment is low. The influence of sediment on the flow field is negligible, thus the simulation of the water and sediment two-phase flow can be simplified as a problem of solving the clear water flow with sediment transport» Dynamic flow equations (general equations of de Saint Venant) A Q + = t x 2 1 Q 1 Q 1 + g I g + β I 2 gs 0 + A t A x A A x A Gravity Local Acceleration Term q l Convective Acceleration Term Pr essure Force Terms Force Term gs = { 0 e Friction Force Term z x y z b, Free surface B(x,η) η h Datum 4 Pressure terms I 1 and I 2 (Cunge et al., 1980) t=time, x=longitudinal coordinate, A=wetted area, Q=flow discharge, g=gravitational acceleration, β=correction factor h h = ( h η) B( x, η) dη I 2 = ( h η) I1 0 0 ( x, η) η B x d? for momentum due to the non-uniformity of streamwise velocity over the cross-section,i 1 =hydrostatic pressure force term, I 2 = a pressure force term that accounts for the forces exerted by the channel wall contractions and expansions, S 0 =longitudinal bed slope, S e =energy slope, q l =lateral flow discharge per unit channel length h= flow depth, B= channel width,η=vertical distance above the channel bottom

5 CLEAR-WATER EQUATIONS Diffusive flow equations The pressure term I 2, and the local and convective acceleration terms in the momentum equation are negligible h S 0 + S e = 0 x The diffusion wave model is more stable than the dynamic wave model But the dynamic wave model can be applied in a wider range of flow conditions Kinematic flow equations The variation in flow depth is negligible in comparison with the variation in channel bed elevation S = 0 S e The kinematic wave assumption is generally applicable if (Dingman, 1984) 5 S gl 0 2 > V 10 V=flow velocity, L=channel length

6 DYNAMIC VS. DIFFUSIVE FLOW EQUATIONS Steady flow through a channel contraction The diffusion wave model exhibits errors in the computed water surface profile in the transition region near the contraction Identical results are obtained in the upstream and downstream regions The relative errors are less than 10%, if the Froude number is less than 0.5, but attains 30% if the Froude number is about 0.97 Wu and Vieira (2002) 6

7 HYDRODYNAMIC FLOW EQUATION FOR WATER-SEDIMENT MIXTURE Any change in flow conditions may be associated with a variation in sediment transport and channel topography, and vice versa The governing equations for clear water equations have limitations In the case of strong sediment transport, the interactions between flow and sediment transport should be taken into the formulation of the water governing equations ( ρa) ( ρq) ( ρ A ) t + x + b t b = ρ 0 2 ( ρq) ρq h 1 ( ρ) t + x A q l + ρga + gah p ρgas 0 + ρgas e x 2 x ( C) C ρ = ρ w 1 + ρ s = 0 A b =cross-sectional area of the bed above a reference datum, ρ=density of the water and sediment mixture in the water column ρ b =density of the water and sediment mixture in the bed surface layer, ρ 0 =density of the water and sediment mixture from tributaries and banks, C=volumetric concentration of total-load sediment, p=porosity of the bed surface layer, h p =mean flow depth ρ b = ρ p + ρ ( 1 p) w s B 2 h p = h ( y) dy / 0 A 7

FLOW RESISTANCE The flow resistance is expressed in terms of the energy slope S e S = e n A 2 2 Q Q R 4 / 3 h n=manning s roughness coefficient, R H =hydraulic radius The Manning n should be

8 FLOW RESISTANCE The flow resistance is expressed in terms of the energy slope S e S = e n A 2 2 Q Q R 4 / 3 h n=manning s roughness coefficient, R H =hydraulic radius The Manning n should be calibrated, but can be evaluated using one of the empirical formulas (e.g., Wu and Wang (1999), van Rin (1984), Karim (1995),Yu and Lim (2003), Recking et al. (2008) ) 1/ 6 d 50 n = Λ Λ log 0.5 g F τ eff T = τ cr, d50 1/ 3 r = log T log Wu and wang (1999) ( T ) n = 0.037d h 50 Karim (1995) F r =Froude number, τ eff =grain shear stress, = bed roughness height, τ cr,50 =critical shear stress for the median diameter d 50, U * =bed shear velocity, ω s =settling velocity Wu (2007) 8

9 COMPARISON OF BED ROUGHNESS FORMULAS Without sediment transport: comparison against 4,376 sets of flume and field data collected by Brownlie (1981) van Rin, Karim, and Wu and Wang formulas almost have the same level of reliability, but work better than Li and Liu formula Wu (2007) With sediment sediment and bed level changes: Saiedi s (1997) experiment Movable bed roughness formulas improves the predicted bed degradation, in comparison with a constant roughness Huang (2007) 9

10 SEDIMENT TRANSPORT EQUATIONS Assumption: the temporal change in relation to sediment discharge is disregarded in the sediment continuity equation Mass conservation by size fraction Q sk x + ( p) A ( AC ) bk k 1 + = t t q sk, l A bk =change in bed area due to grain size class k, Q sk =sediment discharge of the grain size class k, q sk,l =lateral sediment discharge per unit channel length of the grain size class k The amount of sediment in suspension does not change significantly over time as compared to the change of bed elevation, i.e. (AC)/ t<<(1-p) A b / t. With this assumption, the sediment continuity equation becomes the classical Exner equation dq dx sk + A bk ( 1 p) = qsk, l t In some 1-models, the mass balance is expressed as 10 C k t VC + t A k = E k D bk ( 1 p) = Dk Ek t k + q sk, l C k =volumetric concentration of grain size class k, V=flow velocity, E k =entrainment rate of grain size class k from the bed D k =deposition rate of grain size class k onto the bed

11 NON-EQUILIBRIUM SEDIMENT TRANSPORT The sediment continuity equation brings into the modeling problem the volumetric sediment discharge Q s as unknown variable In the equilibrium approach, Q s is assumed equal to the sediment transport capacity Q s cap at every cross-section; Q s cap is given a formula However, the sediment load is generally unable to adapt instantaneously to spatial and/or temporal variations in the flow Non-equilibrium sediment transport accounts for the limited availability of sediment under some special conditions, and the time and space for sediment transport to adapt to its possible capacity in line with the local flow scenario Space-lag equation of Daubert and Lebreton (1969) dq dx sk Q Q cap = sk sk L s = lag distance L s 11

12 LAG DISTANCE PARAMETER The lag distance L s characterizes the distance for sediment transport to reach its saturation rate for a flow condition L s parameter needs to be calibrated, but can be evaluated If there are only ripples on the bed, L s may take the value of the length of ripples (Phillips and Sutherland, 1989), or the average saltation step of particles, which is about 100 times the sediment median diameter (Wu et al., 2000) If dunes are the dominant bed form, L s may take the value length of dune, which is about 7.3 times the flow depth (van Rin, 1984) Bell and Sutherland (1983) found that Ls is equal to time t (in hours); Vieira and Wu (2002) proposed L s = t Rahuel et al. (1989) and Belleudy (2001) link L s to the numerical grid length x and proposed values ranging between one to two times x Han s (1981) formula for suspended sediment: L s = U ω * β 0 s Wu and Wang s (2007) formula: L s Vh = max L b, α 0 ωs L b = lag distance for bed load, α 0, β 0 =suspended load adaptation lengths 12

13 EQUILIBRIUM VS. NON-EQUILIBRIUM SEDIMENT TRANSPORT van Rin (1987) experiment, uniform sediment particles, L s is given by Han s (1981) formula El kadi Abderrezzak and Paquier (2007) 13

14 SEDIMENT TRANSPORT CAPACITY FORMULAS Both equilibrium and nonequilibrium approaches need the computation of the sediment transport capacity formula There is a plethora of formulations derived from differing assumptions and tested with various laboratory and field data Applying different formulas to one problem usually lead to contrasting predictions 14

15 EVOLUTION OF CROSS-SECTIONAL GEOMETRY Cross-sectional shape of a river controls its ability to carry water and to convey sediment, and influences bed shear stresses and thus rate of sediment transport A 1-D model provides only the lumped change in bed area, A b, at a cross-section A b must be allocated appropriately to the local change in bed elevation, z b, along the cross-section at each time step Methods for updating the cross-sectional geometry Uniform distribution of bed change along the cross section Uniform deposition and erosion for wide channels and horizontal deposition and uniform erosion for narrow channels (Wu, 1991) 15

16 16 EVOLUTION OF CROSS-SECTIONAL GEOMETRY A more general method allocates deposition and erosion along the cross-section by a power function of excess shear stress (e.g. Chang (1988), El kadi Abderrezzak and Paquier (2009)) Erosion Deposition x Free surface Datum h B(x,η) η z b, y z x b τ τ m c m c b A y τ τ τ τ z c = >, ) ( ) (,,, Chang (1988), El kadi Abderrezzak and Paquier (2009)) b m c m c b A y τ ξ τ τ ξ τ z = ) ( ) (,,, El kadi Abderrezzak and Paquier (2009)) b m m b A y τ τ z = ) ( ) (, ISIS model, Chang (1988) m=between 0 and 1 according to Chang (1988); depends on the sediment transport capacity used according to El kadi Abderrezzak and Paquier (2008) (=1.5 if Meyer-Peter and Müller formula)

17 FLUVIAL EROSION AT BANK TOES Erosion at bank toes influences channel bed width and bank angle, and causes bank instability The fluvial erosion of cohesive bank material can computed using Arulanandan et al s. (1980) formula db dt = E ρ s g τ τ τ ce ce with r ce ( 0. τ ) = τ exp 13 ce db/dt=lateral erosion rate near the bank toe; τ=flow shear stress; τ ce =critical shear stress for bank toe erosion; E=initial rate of soil erosion The eroded bank material can be treated as lateral sediment transport rate in sediment continuity equation 17

18 BANK FAILURE Channel banks may fail by various mechanisms, which may be (a) rotational (b) planar, (c) cantilever or (d) piping- or sapping-type 18 Langeondoen (2000)

19 BANK PLANAR FAILURE Approach of Simon and Thorne (1988) Factor safety Wu (2007) W t = weight of failure block, C=soil cohesion, φ=soil friction angle, α=angle of bank slope, y d =depth of tension crack, β=angle of failure plane, K tc = ratio of the observed tension crack depth to the bank height Once a mass failure occurs (f s < 1), the volume of the failure block, V f, is 19 The failed material is added in the sediment continuity equation as lateral sediment transport rate, assuming immediate disintegration of the material into its primary particles q sk l = p b k V t f, p k b =fractional content by volume of size class k in the bank

ONE-LAYER AND MULTI LAYER 1-D MODELS Simulation of dam-break flow over movable beds is a challenging issue The sediment concentration is so high and the bed varies so rapidly, the water and sediment

20 ONE-LAYER AND MULTI LAYER 1-D MODELS Simulation of dam-break flow over movable beds is a challenging issue The sediment concentration is so high and the bed varies so rapidly, the water and sediment may form a mixture The sediment transport under dam-break flows is little understood, and the existing sediment transport and flow resistance formulas may not be applicable Various types of 1-D models are used The common Saint-Venant Exner approach using one layer and assuming clear water (e.g. El kadi Abderrezzak and Paquier (2009), Zech et al. (2008)) The basic two-layer model where the velocity of water and sediment layers are considered as unique (e.g. Capart (2000)) The two-layer model in which layer velocities and concentrations are free to be distinct (e.g. Zech et al. (2008)) Water layer Moving sediment layer Fixed bed layer layer 20 After Zech et al. (2008)

21 ONE-LAYER VS. TWO LAYERS MODELS Simulation of a laboratory experiment of dam-break flow over sandy flat bed (Spinwine, 2005) using Zech et al. s 1-D models The one-layer model is progressively delayed The delay in the front movement is still worse in the one velocity/oneconcentration mode Best predictions are given 2-layer 2- velocity/2-oncentration model 21 Zech et al. (2008)

22 ONE-LAYER VS. TWO LAYERS MODELS Simulation of a laboratory experiment of dam-break flow over sandy stepped bed (Spinwine, 2005) using Zech et al. s 1-D model (2-layer 2-velocity/2- oncentration) and El kadi Abderrezzak and Paquier s (2009) model (one layer) Both models provide comparable results, with differences between the predictions of the models being markedly noticed around the initial site of the dam The 2-layer 2-velocity/2-concentration model accelerates the propagation of the wave front 22

23 1-D MODEL FOR WATER-SEDIMENT MIXTURE The classical water and sediment governing equations are written considering a mixture of water and sediment (e.g. model of Wu and Wang (2007)) Simulation of a laboratory experiment of dam-break flow over PVC flat bed using Wu and Wang s 1-D model: the agreement between the simulations and measurements is fairly good 23

24 MAIN ASSUMPTIONS Decoupled Coupled The flow is assumed to be steady when the evolution of the riverbed is studied. Alternatively, the riverbed is implicitly assumed to be fixed within a time step while the flow over the mobile bed is of primary interest In a given time step, the continuity and momentum equations are solved first, assuming negligible bed change rate (i.e. fixed morphology). Then the sediment continuity equation is solved using the flow variables newly obtained The flow and sediment quantities are computed simultaneously Semi-coupled Some quantities are computed in coupled form and the others separately. For example, flow and sediment modules may be decoupled, whereas sediment transport, bed change, and bed material sorting computations may be coupled 24

25 25 JUSTIFICATION OF DECOUPLED, COUPLED AND SEMI COUPLED TECHNIQUES Fully coupled models are more general and sophisticated and may require more computational effort than decoupled models It is claimed that semi-coupled approaches permit arbitrary sediment discharge formulae to be easily incorporated But fully coupled models also allow for an expedient use of arbitrary sediment discharge formulae Use of decoupled approaches may be ustified due to the difference in time scales of flow and sediment transport The time-scale of bed deformation T B, the time-scale of the flow T F and the relative time-scale of bed deformation R T are T B =h/(cω s ) T F =h/v R t =T B /T F = V/(Cω s ) The product Cω s is low compared to the flow velocity V R T >>1 But there has been no quantitative measure of R T for decoupling to be acceptable

26 DECOUPLED VS. COUPLED TECHNIQUES Aggradation case (Cao et al., 2002) In the upstream half of the channel, there exists evident undulation, attributed to the different physical and numerical background of the fully coupled model (FCM) and and decoupled model (DCM) Degradation case (Cao et al., 2002) In the upstream part, the discrepancy in changes of bed elevation is ascribed to the inherent ignoring of bed mobility and the asynchronous solution procedure of the DCM 26

27 DATA REQUIREMENTS Study domain: Inlet and outlet of the channel under interest should be located near gauge stations or control structures where measured flow and sediment data are available Channel topography If hydraulic structures are involved, their geometries should be provided Manning roughness coefficient The Manning n should be estimated using measured flow data. Empirical formulas may be used if no measurement data are available. n in streams with similar flow and sediment conditions may be used as reference Sediment particle properties Covering all sizes of bed load, suspended load, bed material, and bank material The entire size range is divided into a suitable number of size classes 27

28 DATA REQUIREMENTS Bank-material properties Density and grain size composition. For a cohesive bank: cohesion, friction angle of the bank material, critical shear stress for bank toe erosion Boundary conditions Inflow water discharge, water stage (at inlet or outlet depending on the flow regime), inflow sediment discharge and size composition Historical data Historical measurement data of flow, sediment discharges, channel morphological changes, etc., should be collected and analyzed for better understanding of the study problem and calibration of the numerical model 28

29 SELECTED 1-D MORPHODYNAMIC MODELS After Papanicolaou et al. (2008) Model and references Flow Bed sediment transport Suspended sediment transport Sediment mixtures Cohesive sediment Source Executable code Language HEC-6 (Thomas and Prashum, 1977) MOBED: MObile BED (Krishnappan, 198)1 Steady Yes Yes Yes No PD PD F77 Unsteady Yes Yes Yes No C C F90 IALLUVIAL (Karim and Kennedy, 1982) Quasi- Steady Yes Yes Yes No C C FIV FLUVIAL 11 (Chang, 1984) Unsteady Yes Yes Yes No C C FIV GSTARS (Molinas and Yang, 1986), now named SRH-1D CHARIMA (Holly et al., 1990) SEDICOUP (Holly and Rahuel, 1990) OTIS (Runkel and Broshears, 1991) Unsteady Yes Yes Yes No PD PD F90/95 Unsteady Yes Yes Yes Yes C C F77 Unsteady Yes Yes Yes No C C F77 Unsteady No No No No PD PD F77 EFDC1D (Hamrick, 2001) Unsteady Yes Yes Yes Yes PD PD F77 CONCEPTS (Langendoen et al., 2001) CCHE1D (Vieira and Wu, 2002) 3STD1 (Papanicolaou et al., 2004) RubarBE (El kadi Abderrezzak and Paquier, 2009) Unsteady Yes Yes a Yes Yes PD C? Unsteady Yes Yes Yes No PD C F77/90/C Unsteady Yes a Yes a Yes No C C F90 Unsteady Yes a Yes a Yes No C P F77 29 Note: C:copyrighted; P:proprietary; PD:public domain; F:FORTRAN. a : Treated as a total load without separation

30 1-D MORPHODYNAMIC MODELS TO DOWNLOAD CCHE1D: CONCEPTS: SRH-1D (GSTARS): 30

BED LOAD SEDIMENT TRANSPORT

BED LOAD SEDIMENT TRANSPORT Kamal EL KADI ABDERREZZAK EDF-R&D, Laboratoire National d Hydraulique et Environnement (LNHE) 1 17-19 September 2009 UNL, Santa Fe, Argentina OUTLINE I. Bed load II. Settling