Journal of Hydraulic Research Vol. 00, No. 0 (2004), pp. 1–19 © 2004 International Association of Hydraulic Engineering and Research
One-dimensional hydrodynamic/sediment transport model applicable to steep mountain streams Modèle Unidimensionnel hydrodynamique/ transport de sediment applicable aux torrents rapides de montagne A. N. PAPANICOLAOU, Associate Professor, Department of Civil and Environmental Engineering, IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, IA 52242, USA. E-mail:
[email protected] (author for correspondence) A. BDOUR, Assistant Professor, Department of Civil and Environmental Engineering, The Hashemite University, Zerqa, Jordan. (Formerly: Graduate Research Associate, Department of Civil and Environmental Engineering, Washington State University, P.O. Box 642910, Pullman, WA 99164-2910, USA) E. WICKLEIN, Engineer ENSR, Redmond, WA 98052-3422, USA. (Formerly: Graduate Research Associate, Department of Civil and Environmental Engineering, Washington State University, P.O. Box 642910, Pullman, WA 99164-2910 USA) ABSTRACT A new one-dimensional (1-D) numerical model for calculating flow and sediment transport in steep mountain streams is developed. 3ST1D, which stands for Steep Stream Sediment Transport 1-D model, is applicable to unsteady flow conditions that occur over transcritical flow stream reaches such as flows over step-pool sequences. 3ST1D consists of two coupled components, the hydrodynamic and the sediment transport. The flow component is addressed here by solving the unsteady form of the Saint-Venant equations. The Total Variation Diminishing Dissipation (TVD)-MacCormack scheme, which is a shock-capturing scheme capable of rendering the solution oscillation free, is employed here to approximate the hydrodynamic solution over transcritical flow stream reaches. The sediment component of the model accounts for multifractional sediment transport and incorporates a series of various incipient motion criteria and frictional formulas applicable to mountain streams. In addition, sediment entrainability is estimated based on a state-of-the art formula that accounts for the bed porosity, turbulent bursting frequency, probability of occurrence of strong episodic turbulent events, and sediment availability in the unit bed area. The model at the end of each time step predicts the flow depth, velocity and shear stress distribution within a cell and calculates changes in bed evolution and grain size distribution. The overall performance of the model is evaluated by comparing its predictions with observations from two flume studies, two field investigations and against the predictions of the quasi-steady model of Lopez and Falcon developed for mountain streams. A sensitivity analysis is performed to assess the effects of cell size and Manning’s roughness coefficient in the predictive ability of the model. RÉSUMÉ Un nouveau modèle numérique uni-dimensionnel (1-D) de calcul d’écoulement et de transport de sédiment dans les torrents rapides de montagne a été développé: 3ST1D, mis pour Steep Stream Sediment Transport 1-D model (modèle de transport 1-D de sédiment dans les torrents à forte pente), est applicable aux écoulements instationnaires qui se produisent dans les biefs critiques tels que les séquences de marches. 3ST1D se compose de deux composantes couplées, l’hydrodynamique et le transport de sédiment. Pour la composante écoulement on a recours à la forme instationnaire des équations de Saint-Venant. Le schéma TVD de MacCormack (diminution de la variation totale de dissipation) qui est un schéma absorbant les chocs capable de rendre la solution libre d’oscillations , est utilisé pour approcher la solution hydrodynamique dans les biefs du torrent où l’écoulement est transcritique. La composante de sédiment du modèle tient compte de sédiments composites et incorpore une série de critères pour le début d’entraînement et de formules de frottement applicables aux torrents de montagne. En outre, l’entraînement de sédiment est basé sur l’état de l’art qui tient compte de la porosité de lit, de la fréquence des bouffées turbulentes, de la probabilité d’occurrence d’événements turbulents épisodiques forts, et de la mobilité des sédiments dans le secteur de lit. Le modèle à la fin de chaque pas de temps donne la profondeur, la distribution de vitesse et de contraintes de cisaillement dans une cellule et calcule l’évolution du lit et de la distribution des tailles de grains. La performance globale du modèle est évaluée en comparant ses prévisions aux observations de deux études en canal, de deux expériences en nature et des prévisions du modèle quasi-stationnaire de Lopez et de Falcon développé pour des torrents de montagne. Une analyse de sensibilité est exécutée pour évaluer les effets de la taille de maille et du coefficient de rugosité de Manning sur les capacités prédictives du modèle.
Keywords: 1-D unsteady sediment transport model, step-pool sequences, steep mountain streams, turbulent bursts. 1 Introduction
or more (Bathurst et al., 1987). The bed morphology of mountain streams can be classified as pool-riffle, coarse-riffle, plane configuration, step-pool, chute, or waterfall (Padmore et al., 1996; Wohl and Thompson, 2000). Such classifications are correlated with
Steep mountain streams are typically characterized as streams having longitudinal gradients ranging from 0.1% to almost 20% Revision revised 1
2
Papanicolaou et al.
the local stream gradient, with pool-riffle sequences observed at slopes less than 2%, and coarse-riffles or cascades at slopes ranging from 3 to 7% (Grant et al., 1990). The bed often goes to a plane configuration before ending in a series of step-pools, as the slope progressively increases from 7 to 15% (Billi et al., 1998). Beyond slopes of 15%, the dominant stream configuration is chutes or waterfalls (Montgomery and Buffington, 1993). As these bed configurations develop, flow in many reaches of the stream can become supercritical for at least some of the annual discharge (Glacey and Williams, 1994). Physical and/or numerical simulation of flow and sediment transport in mountain streams is hindered by the presence of high gradients, large roughness elements, high turbulence and variable sediment availability (Wohl and Thompson, 2000). The bed material of such streams is characterized of variable bed roughness and it is a mixture of sand, gravel, cobbles, and boulders up to 1–2 m in diameter (Trieste, 1992). Flow resistance is high and energy losses increase due to the presence of variable bed topography, wake turbulence and localized hydraulic jumps that are formed behind protruding resistance elements (Bayazit, 1982). The interaction of flow and bed roughness elements leads to the genesis of a metastable cycle of turbulent eddies known as bursts (the cycle of sweeps, ejections, inward and outward interactions) which are considered responsible for the commencement of sediment motion and transport of sediment (Papanicolaou et al., 1999). Characterization of the near-bed turbulence and thereby the sediment transport in such flows generally requires more information than the local average bed shear stress (Nelson et al., 1991). Another major difficulty in replicating the interaction of flow and sediment in mountain streams is lack of accurate methods for predicting flow resistance (Thorne and Zevenbergen, 1985). Most of the conventional methods for predicting flow resistance focus on grain (skin) resistance neglecting the effects of form (Rice et al., 1998; Maxwell and Papanicolaou, 2001; MacFarlane and Wohl, 2003). The limitations of these methods are quite significant in streams where the vertical protrusion of the largest particles is relative large and sometimes exceeds the bankfull depth of flow and also in streams with ubiquitous macro- and micro-bed features (Millar, 1999). When the relative submergence (the ratio of depth with the median particle diameter) is less than 5, the assumptions used to derive the law of the wall equation are violated and the flow velocity does not follow a logarithmic profile (Nelson et al., 1991). In addition, if the particles break the water surface, a wave drag will be exerted on the fluid which will alter the reach-averaged flow properties (Schlichting, 1979). Along the same lines, quantifying the critical stress of the sediment particles in mountain streams poses an extra degree of difficulty. According to Wittler and Abt (1995), the incipient motion of sediments is affected by the surface waves and the entrained air bubbles that are generated as the flow plunges to the protruding roughness elements; in this case, the critical stress becomes strongly dependent of the densimetric Froude number instead of the particles Reynolds number (Aguirre-Pe et al., 2003). Other researchers (e.g. Suszka, 1991) suggest that the critical stress relates to the relative submergence. For many years, the flow and sediment motion in steep mountain streams have been investigated via physical models
(e.g. Whittaker, 1987). However, this method is generally very time consuming and for many projects is impractical due to the high cost associated with the construction of physical models. Hence, there is a strong need for numerical models that predict the temporal evolution of mountain streams under different flow conditions and sediment availability (Lenzi, 2001; Rathburn and Wohl, 2001). In recent decades, several semi-coupled and fully coupled hydrodynamic/sediment transport models have been developed for commercial and research engineering projects. However, very few of these models are applicable to steep mountain streams. The studies of Bradley et al. (1998) and Miller and Cluer (1999) provide a comprehensive review of published modeling studies. Known one-dimensional (1-D) models such as HEC-6, HEC-14, MIDAS, GSTARS3.0, FLUVIAL12, FLDWAV, SEDROUT are mainly applicable to lowland alluvial rivers. Along the same lines, 2-D hydrodynamic/sediment transport models such as the SMS family of models (RMA2, FESWMS), HIVEL2D, CCHE2D, SEDZL are mainly developed for estuarine or large riverine environments where low gradient and clay/sand uniform mixture of sediments are the dominant bed materials (Pavlovic et al., 1985; Thomas et al., 1985; Spasojevic and Forrest, 1990). Finally, most of the 3-D models are strictly hydrodynamic models (e.g. FLUENT, FLOW3D) while the few 3-D models that couple the flow component with the sediment are only applicable to sand bed streams (FAST3D, CH3D-SED). The models of Li and Fullerton (1987), Pianese and Rossi (1991), Di Silvio (1992), and Lopez and Falcon (1999) are one of the few existing models that predict sediment routing in steep mountain streams. A common characteristic of these models is that they are 1-D and treat the flow as steady and uniform and simulate changes in the composition of sediment sizes within the bed by imposing continuity for individual size fractions. The Lopez and Falcon (1999) model offers a robust technique in determining bed changes in mountain streams by considering two layers in the bed: an upper layer where the interaction of flow and sediment occurs and an underlying (parent) layer that provides material to the upper layer as surficial erosion evolves within a stream reach. The drawback of the Lopez and Falcon model is that it treats flow as subcritical by ignoring the transcritical and unsteady nature of the flow. Although this consideration maybe true for mountain streams with plane bed morphology, it is not valid for flows over bed configurations such as pool-riffle and step-pool sequences, sediment bars and cluster microforms (e.g. Reid et al., 1992; Nicholas, 2001). It is therefore imperative that future models will account for the presence of transcritical flow conditions and high level of turbulence (Trieste, 1992; Glacey and Williams, 1994). A prerequisite for such a development is the employment of numerical schemes than can simultaneously handle calculations of slowly varying flows as well as rapidly varying ones containing shocks or discontinuities without generating the spurious oscillations that most classical methods do. The objective of this research is to develop a steep mountain stream model that removes some of the limitations of the existing models and accounts for the complex interaction of flow, geomorphology and sediment transport. The model proposed herein, is limited to 1-D flows and consists of a hydrodynamic
1-D hydrodynamic/sediment transport model
component and a sediment transport component which are solved simultaneously. 3ST1D, which stands for Steep Stream Sediment Transport 1-D model, is applied for the first time to unsteady flow conditions occurring over transcritical flow regimes such as flows over step-pool sequences. The model can handle abrupt changes in flow (transcritical flows), the presence of multifractional bed size distribution, and incorporates the episodic role of turbulence on sediment erosion. The flow is calculated by solving the unsteady form of the Saint-Venant equations. The TVDMacCormack scheme, which is a shock-capturing scheme with total variance diminishing, is employed here to approximate the hydrodynamic solution. Its main advantage is that it can handle flows containing shocks or discontinuities such as flows over steppool sequences or flows around the vicinity of large roughness elements. The sediment component of the model accounts for multifractional sediment transport and incorporates a series of various incipient motion criteria and frictional formulas applicable to mountain streams. The stream sediment-capacity is predicted by the Schoklitsch method, which adequately predicts bedload motion in streams under transport- and supply-limited conditions (Lopes et al., 2001). In addition, sediment entrainability is estimated based on a state-of-the art formula that accounts for the bed porosity, bursting frequency, probability of occurrence of strong episodic turbulent events, and sediment availability in the unit bed area. The model at the end of each time step predicts the flow depth, velocity and shear stress distribution within a cell and calculates changes in bed evolution and grain size distribution. The overall performance of the model is evaluated by comparing its predictions with observations from two flume studies, two field investigations and against the predictions of the quasi-steady model of Lopez and Falcon developed for mountain streams. A sensitivity analysis is performed to assess the effects of Manning’s roughness coefficient and cell size in the predictive ability of the model. The remainder of the paper is organized as follows: first the hypotheses used to develop the hydrodynamic component are presented; second a detailed description of the TVDMacCormack scheme is provided along with the initial and boundary conditions. Third, a description of the components of the sediment transport model is provided. This includes the flow resistance formulas, the sediment capacity equation, the erosion equation, and the formulation used to update the sediment bed. The results of the model, comparison with the literature field data and a sensitivity analysis are discussed at the last part of the manuscript.
2 Hydrodynamic model 2.1 Model equations 3ST1D model is developed based on the following considerations: 1. Flow in mountain streams is predominantly unidirectional and therefore, the temporal and spatial characteristics of flow can be adequately described by using the unsteady form of the
3
1-D Saint-Venant equations (van Niekerk et al., 1992). The 1-D Saint-Venant equations express the law of conservation of mass and momentum along the flow direction (Cunge et al., 1980). 2. The mountain streams can be adequately simulated as channels with prismatic cross-sections, in cases that channel cross-sectional measurements are not available (Lopez and Falcon, 1999). 3. Friction at the bed of the channel is dominant over internal shear stresses, and thus the latter can be considered negligible (Brufau et al., 2000). Under these assumptions, the Saint-Venant equations can be written in conservative form as follows: ∂U ∂F + =G (1) ∂t ∂x where A U= (2a) Q Q F = Q2 (2b) + gI1 A 0 G= (2c) gI2 + gA(S0 − Sf ) and A[x, H (x, t)] is the wetted cross-section area; H (x, t) is the water depth; Q(x, t) is the discharge; g is the acceleration of gravity; and I1 is the hydrostatic pressure force term that can be written as H (x,t) I1 = [H − η]σ (x, η)dη (3) 0
with σ σ (x, η) =
∂A(x, η) ∂η
(4)
This represents the channel width at the free surface for a water depth η. The term I2 appearing in the source term is defined as follows: H (x,t) ∂σ (x, η) I2 = dη (5) [H − η] ∂x 0 Equation (5) accounts for the forces exerted by the channel walls contractions and expansions by including the gradient of σ with respect to x; S0 and Sf are the bottom and friction gradients, respectively. The conservative form of equation guarantees correct jump intensities and celerities of surface waves (Lax and Wendoff, 1960). The TVD-MacCormack scheme is employed here to approximate the solution of the Saint-Venant equations (Garcia-Navarro et al., 1992). The TVD-MacCormack scheme is an expansion of the widely used MacCormack scheme (MacCormack, 1969) and it includes a shock-capturing method with second order of accuracy, capable of rendering the solution oscillation free without introducing any additional difficulty for the treatment of the source term G. The algorithm procedure involves a two-step procedure due to the presence of the source term G, which is known as the “predictor–corrector” algorithm (Garcia-Navarro et al., 1992).
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Papanicolaou et al.
The predictor terms for the depth H or velocity U using the backward finite difference scheme for Eqs (1) and (2) can be written, respectively, as follows: t j j j j j j j j H˜ i = Hi − Hi Ui − Hi−1 + Ui Hi − Hi−1 x
j 2 j 2 Ui − Ui−1 t j j j j U˜ i = Ui − g + Hi − Hi−1 x 2g j 2 n2 Ui + gt S0 − j Ri
(6)
(7)
j j where H˜ i and U˜ i are the values of depth and velocity, respectively, at the end of the predictor part, i and j denote space, referred here as cell and time, respectively. t and x denote the time step and cell space between two subsequent nodes, respectively. The coefficient n in Eq. (7) denotes the Manning’s j resistance coefficient and Ri denotes the hydraulic radius. The corrector Eqs of (1) and (2) are expressed using the forward finite difference scheme as follows:
t j ˜ j j j j j Hi Ui+1 − U˜ i + U˜ i H˜ i+1 − H˜ i x
j 2 j 2 ≈j U˜ i+1 − U˜ i t j j j + H˜ i+1 − H˜ i U i = U˜ i − g x 2g j 2 n2 Ui + g S0 − j Ri ≈j
j
H i = Hi −
≈j
(8)
(9)
≈j
where H i and U i are, respectively, the corrected values of depth and velocity at the end of the corrector step. At the end of a time step calculation, the numerical solution of depth and velocity is calculated by averaging the predicted and corrected values, determined earlier, and by adding the TVD dissipation term E to provide an oscillation free solution. Therefore, j +1
Hi
=
j ≈j 1 ˜ j j + λ Ei+1/2 − Ei−1/2 Hi + H i 2 predictor–corrector terms
j +1 Ui
(10)
k k 1 k 1 − λa¯ i+1/2 αi+1/2 ψ a¯ i+1/2 2 k=1 k × 1 − φ r¯i+1/2
The function ψ prevents the appearance of unphysical hydraulic jumps; i.e., those in which energy increases across the shock, which are admissible for the classical MacCormak scheme (Yee, 1989). Finally, the factor φ in Eq. (12) is a limiter parameter responsible for obtaining non-oscillatory solutions despite the presence of strong gradients or shocks. It is a nonlinear function of the ratio: k ri+1/2 =
k αi+1/2−s k αi+1/2
k s = sign(ai+1/2 )
The MacCormack scheme must satisfy the Courant–Friedrich– Lewy (CFL) criterion at each cell in order to be stable. The CLF criterion is defined: t Cni = (13) U + gH ≤ 1 x where Cni is the Courant number at cell i. The value of Cn , in general, must be less than or equal to 1. If the source term G in Eq. (1) obtains significant values it is recommended that the Cn value be adjusted to values smaller than 1 in order to maintain model stability (Brufau et al., 2000).
(11)
2
k = a discrete average characteristic velocity of the a¯ i+1/2 states at i and i + 1, expressed as k a¯ i+1/2 = u¯ i+1/2 ± c¯i+1/2
where ε is a small positive number whose value has to be determined for each individual problem. Harten and Hyman (1983) suggested a formula to calculate ε in order to cut down the trial process. k k εi+1/2 = max 0, a¯ i+1/2 − a¯ ik
3 Sediment transport model
where the E term is defined as j
k The function ψ in Eq. (12) is an entropy correction to a¯ i+1/2 , and in its simplest expression takes the form k k k ≥ εk ψ a¯ i+1/2 = a¯ i+1/2 if a¯ i+1/2 i+1/2 k k k k ψ a¯ i+1/2 = εi+1/2 if a¯ i+1/2 < εi+1/2
TVD dissipation term
j ≈j 1 j j = U˜ i + U i + λ Ei+1/2 − Ei−1/2 2
Ei+1/2 =
where the symbol (− ) denotes time-averaging and the quantities u¯ i+1/2 and c¯i+1/2 = discrete approximations to the local water velocity and wave celerity, respectively. √ √ (Qi+1 / Ai+1 ) + (Qi / Ai ) u¯ i+1/2 = √ √ Ai+1 + Ai ci+1 + ci c¯i+1/2 = 2
(12)
The sediment transport model is composed of several interconnected components. These components include: determination of the total and grain resistance; determination of the incipient motion conditions in high gradient streams; calculation of the stream sediment-carrying capacity; consideration of turbulent bursts and inclusion of the turbulent fluctuations in estimating sediment erosion; changes in bed elevation and temporal evolution of the surface grain size distribution.
3.1 Resistance to flow The majority of the formulas found in the literature provide only the resistance due to skin friction (grain roughness) while few
1-D hydrodynamic/sediment transport model
formulas account for form resistance or for the total resistance (Thorne and Zevenbergen, 1985). In the present model, the total resistance f = f + f , where f = grain resistance and f = form resistance, is estimated for the first time by employing the semiempirical formula of Maxwell and Papanicolaou (2001). This formula is applicable to steep mountain streams with well defined step-pool bedform configurations and is described as follows: dstep d84 8 = −3.73 log − 0.80 (14) f LH where dstep is the average step height; d84 is the size of sediment for which 84% of the sample is finer; and L is the average step length. In the absence of step-pool bedform configurations, grain resistance is estimated by using different semiempirical formulas depending on the stream gradient. For slopes within the range of 0.4–4%, we employ the Bathurst’s (1985) formula: 1/2 8 H +4 (15a) = 5.62 log f d84 or the equation developed by Aguirre-Pe and Feuntes (1993) for low relative submergence (H /d50 ): 1/2 8 d50 H + 3.09 + 1.75 = 2.5 ln (15b) f d50 H For slopes varying within the range of 4–30%, the Abt et al. (1987) and Rice et al. (1998) formulas are utilized. The Abt et al. (1987) study provides a relationship for the Manning’s resistance coefficient: n = 0.0456 (d50 S)0.159 f
(16)
where d50 is the size of sediment calculated in inches for which 50% of the sample is finer. The Rice et al. (1998) formula is: 1/2 8 H +6 (17) = 5.10 log f d84 3.2 Incipient motion Accurate prediction of sediment transport requires the establishment of rigorous criteria for the commencement of sediment motion. The traditional Shields (1936) criterion is clearly inappropriate for use in mountains streams due to the lack of plane bed configurations and sediment size uniformity. In particular, the Shields relation does not account for hiding effects and relative protrusion of particles, which are two of the most common features in mountain stream with non-uniform bed material. In addition, in steep mountain streams the particles Reynolds number seizes to be the governing parameter on the commencement of sediment motion and the initiation of motion of particles becomes strongly dependent of the Froude number and/or relative submergence (Bettess, 1984; Wittler and Abt, 1995). The modification of the Shields criterion to account for the presence of hiding effects and relative protrusion resulted to the development of several semiempirical and analytical approaches for determining the incipient motion conditions in mountain
5
streams (Kannellopoulos and Diplas, 1998). The most prominent ones relate the dimensionless critical shear stress, known as the Shields parameter, with the relative submergence (e.g. Bettess, 1984; Suzka, 1991). Other approaches provide relations between the Shields parameter and the Froude number (e.g. Kilgore and Young, 1993; Grant, 1997). Aguirre-Pe and Fuentes (1993), among others, realized the importance of the Froude number and relative submergence in the commencement of sediment motion and they developed formulas that clearly indicate their strong interdependence. 3ST1D takes into account the latest developments in the incipient motion theory and allows the selection of different incipient motion criteria depending on the stream gradient. The criteria considered here are: 1. The Suszka (1991) criterion, which is applicable to streams with slopes varying within the range of 0.2–9%. The Suszka (1991) formula developed for a d50 range of 3–44 mm relates the dimensionless critical shear stress of the d50 sizedsediment, τcr∗ 50 , with the relative submergence defined here as the ratio H /d50 : τcr∗ 50 = 0.0851
H d50
−0.266 (18)
When the sediment bed layer is represented by a multifractional sediment distribution, Eq. (18) is modified to account for the hiding effects introduced by the larger size fractions. The dimensionless critical shear stress for the d50 sizedparticles, τcr∗ 50 , is fractionalized for all classes based on the following relation: τcr∗ pr
=
τcr∗ 50
¯ −m dpr d50
(19)
where d¯pr is the median particle diameter of a fraction with size p and density r and τcr∗ pr is the dimensionless critical shear stress for that fraction. The coefficient m in Eq. (19) is indicative of the selective transport of sediment particles due to hiding effects. Values of m < 1 indicate that the finer particles are mobilized at critical shear stresses smaller than those for coarser particles, whereas when m > 1, the coarse particles are mobilized first. Komar (1989) suggested that a value of m = 0.65 is appropriate to use in Eq. (19) and this value is adapted here. 2. The Graf and Suszka (1987) criterion, which is developed for the same d50 range with the Suszka formula, is applicable to higher gradient streams, viz. with slopes varying within the range of 2–20%. The Graf and Suszka criterion suggests that τcr∗ 50 = 0.042(102.2S0 )
(20)
In the presence of a multifraction sediment distribution, the dimensionless critical shear stress for the d50 sized-particles, τcr∗ 50 , in Eq. (20) is fractionalized for all classes based on Eq. (19).
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Papanicolaou et al.
3.3 Sediment-carrying capacity For a given sediment composition, a certain flow can only carry a certain sediment capacity without net deposition and erosion. This quantity is called the sediment-carrying capacity (Chang, 1988). In 3ST1D the Schoklitch (1962) equation is employed to determine the stream carrying sediment capacity since it is reported in the literature as the best predictor equation for bedload transport in mountain streams for various sediment availability conditions, namely, Transport Limited condition 1 (TL1), Transport Limited condition 2 (TL2), and a supply Limited Condition (SL) (Lopes et al., 2001). TL1 represents a condition that sediment availability of all sizes is present; TL2 represents a condition that sediment availability of some sizes is available; and SL condition occurs when sediment availability of all sizes is not present. In addition, the Schoklitch equation has the distinct advantage over other equations in that it is a function of flow, and not depth or width, both parameters that are very difficult to measure in the field (Bathurst et al., 1987). The Schoklitch equation is given by the following expression: qsb =
2.5 3/2 S (q − qc ) ρs /ρ f
(21)
where qsb is the volumetric bedload transport rate per unit width or sediment-carrying capacity, q the flow discharge per unit width, ρs the density of the sediment, ρ the fluid density, and qc the critical discharge that is described as follows: 3 (22) qc = 0.21Sf−1.12 gd16 where d16 is the size of sediment for which 16% of the sample is finer.
3.4 Turbulent bursts and sediment erosion The presence of small-scale turbulent coherent structures near a boundary layer leads to a phenomenon known as the turbulent bursting process, which appears to be closely tied to the commencement of sediment motion and subsequently to bed erosion process in natural riverine systems (Bridge and Best, 1988; Niño and Garcia, 1996; Cao, 1997; Papanicolaou et al., 2001). The turbulent bursting process is characterized by a metastable cycle of ejections, sweeps, inward and outward interactions (Kline et al., 1967). Bursts are believed to have distinct mean (timeaveraged) spatial and temporal characteristics and they are found to affect the magnitude of the instantaneous bed shear stress yielding instantaneous bed shear stress values that are in excess of two to six times the mean (Grass, 1971; Nezu and Nakagawa, 1993). The incorporation, therefore, of the mean spatial and temporal characteristics of turbulent bursts in sediment transport modeling is of paramount importance in predicting adequately the commencement of sediment motion and modeling sediment erosion. The spatial characteristics of bursts for smooth beds are described by a time-averaged burst area. As a first approximation, it is estimated from measurements (Kline et al., 1967; Kim et al.,
1971; Rao et al., 1971; Wallace et al., 1972) that turbulent bursts extend, on average, for about lx = 40ν/U∗ in the streamwise (x-direction) and ly = 25ν/U∗ in the transverse (y-direction) directions. The bursts are distributed with a spacing on nearly Sx = 500ν/U∗ longitudinally and Sy = 100ν/U∗ in the transverse direction, where ν is the kinematic viscosity and U∗ the √ shear velocity defined as U∗ = gH Sf . Then the time-averaged area of a burst is given by the following relation: l=
lx ly 40 × 25 = 0.02 = S x Sy 500 × 100
(23)
For rough beds it is determined that the average area AB of the bursts scales with the size of bed roughness (Grass et al., 1993). It is, therefore, expected that AB should be greater than l (Grass et al., 1993). Asaeda et al. (1989) studied the entrainment of sediment in alluvial environments and found that the fraction of the bed area from which sediment erosion occurs, increases proportionally to the local bed shear stress. Following this qualitatively, it is hereby considered that the normalized area of a burst, which is formed within a stream reach with a bed area Abed , can be approximated as, AN =
lτN τcr∗ pr
(24)
where AN = AB /Abed and τN = τ/τ ·τ is the instantaneous bed shear stress equal to τ +τ ; τ is the time-averaged component of the bed shear stress equal to (f /4)(ρU 2 /2); τ = fluctuating component of the instantaneous effective bed shear stress. The approximation in (24) lends a limitation, τN /τcr∗ pr ≤ 50, to the erosion model because quantity AN cannot exceed 100%. The normalized instantaneous stress τN is approximated with a basic Gamma distribution with a probability density function fτN (Paola, 1996; Papanicolaou and Hilldale, 2002), fτN = α α τNα−1
e(−ατN )
(α)
(25)
where α is a non-dimensional measurement of the distribution width, equal to the inverse square of the coefficient of variation (standard deviation/mean) and (α) is the Gamma function of α,
(α) = (a − 1)! α is taken as four following the recommendation of Papanicolaou (1997) for streams with isolated roughness elements. The temporal characteristics of bursts are described by an average bursting period which is scaled for rough beds with the outer flow variables, namely, average flow depth and depth-averaged velocity (Cantwell, 1981), TB = (3 − 6)
H U
(26)
The rate of mass erosion within a stream reach with bed area Abed for a time step t is predicted using the spatial and temporal characteristics of bursts and the concept of probability of entrainment (Jain, 1992). The probability of entrainment PE is equal to the proportion of time that the normalized instantaneous
1-D hydrodynamic/sediment transport model
stress τN exceeds τcr∗ pr . The rate of erosion is strongly dependent of the active layer thickness DA (active layer is defined as the layer where sediment activity in the form of erosion or deposition occurs), porosity θ , and Fpr which is the fraction of particles with size dpr present in the active layer, and is described by the following relation (Jain, 1992), E = ρs AN DA (1 − θ )Abed Fpr
PE t TB TB
3.5 Treatment of the bed layer The 3ST1D domain consists of cells with three-layers as it is illustrated in Fig. 1. The first layer refers to the flow region (water column) where sediment is routed downstream with the water flow. The second layer is the active layer with thickness DA where sediment is exchanged between the bed and the flow. The third layer is the parent layer to the active layer and it is considered here as infinitely deep. The parent layer supplies sediment to the active layer when the sediment composition of the second layer is updated at the end of each time step t. The treatment of the bed, within a cell i, includes the following steps: (1) estimation of the fraction of the material found within the bed and modification of the carrying-capacity equation (21) to account for multifractional transport; (2) determination of the
(27)
where
∗ τcr pr
PE = 1 −
fτN dτN
0
The active thickness DA is typically set equal to dmax (Park and Jain, 1987) or equal to 2dmax (Diplas and Fripp, 1992).
(a)
Node 1
Node 2
Control Volume
z x
Node i-1
H Flow Region
Node i
Node i+1 Node End
U i-1
Active Layer
Ui
U i+1
Parent Layer
x
(b) Cell Length
Flow Volume Flow Out
Volume Flow In Flow Depth
7
Volume Deposition and Scour and Suspension
Active Layer Thickness
Parent Layer
Figure 1 (a) A sketch of the domain of the model. (b) A description of a cell and definition of the active layer.
8
Papanicolaou et al.
scour and depositional depths; and (3) update of the thickness of the active layer after deposition or scour occurs. The estimation of the fraction of the material found within the bed requires the following steps to be taken: (1) determination of the flow velocity Ui via Eq. (10); (2) determination of the travel time ti of sediment through the cell i. The travel time is ti = xi /Ui . The travel time ti must always be greater than or equal to the computational time step t; the condition t ≥ ti ensures sufficient time for sediments to route through a cell; and (3) determination of the time fraction (TFRAC )i = t/ti , which is ultimately incorporated into the formulation for predicting the carrying capacity. The boundary conditions for the 3ST1D hydrodynamic component are the volumetric flow discharge at the upstream end of the domain and the flow depth at the downstream end. For the sediment transport component of the model, the percentage Fpr of the available particles with size class p and density r at time j is provided. The volume of the available material within the active layer thickness DA of the cell i is estimated as j
j
(∀ACTpr )i = (DA Fpr (1 − θ )Abed )i
(28)
If ∀INpr denotes the volume of sediment particles with size class p and density r entering the cell, then the total available volume of sediments with size class p and density r at time j is given as j
j
(∀FRACTIONpr )i = (∀ACTpr + ∀INpr )i
(29)
and the total volume for all fractions available in the cell bed will be j N,M N,M j (∀FRACTION_TOTAL )i = ∀ACTpr + ∀INpr p=1,r=1
p=1,r=1
i
(30) where N is the maximum number of sediment size classes present in the mixture and M the maximum number of sediment density classes present in the mixture. The sediment carrying-capacity equation, (21), is modified to account for the presence of multifractional sediment and the travel time of sediments as follows j ∀FRACTIONpr 2.5 3/2 j (qsbpr )i = TFRAC Sf (q − qc ) ρs /ρ ∀FRACTION_TOTAL i (31) In terms of units of volume, the sediment carrying-capacity is j j written as (∀sbpr )i = (qsbpr Bt)i where B is the width of the cell. Within the cell i different sediment class sizes and densities will deposit or erode depending on the existing flow conditions, sediment class incipient conditions, bed roughness and level of turbulence. To determine if scour or deposition will occur for a given sediment class p, r within the cell i at time j , the volume of j sediment-carrying capacity (∀sbpr )i is compared to the volume j of the inflow sediment particles (∀INpr )i of the same class. If the transport capacity exceeds the volume inflow for the given class, then that class is scoured from the bed until transport meets
capacity. The volume of scour within the cell i for the sediment class p, r, which is determined via Eq. (27), is expressed here as j j (∀SCpr )i = (Epr t/(ρs )r )i and the scour depth is determined as j j (SCpr )i = (∀SCpr /Bx)i . The volume of sediment with class j j j p, r that exits the cell is (∀OUTpr )i = (∀SCpr )i + (∀INpr )i . If the transport capacity is less than the volume inflow for the sediment class p, r then deposition occurs. The volume of j j j deposition is (∀DPpr )i = (∀INpr )i − (∀sbpr )i and the volume j of sediment with class p, r that exits the cell is (∀OUTpr )i = j j (∀INpr )i − (∀DPpr )i . The deposition depth is determined as j j (DPpr )i = (∀DPpr /Bx)i . For all cases, it is verified that the continuity equation is satisfied within the cell i for the sediment j j j class p, r at time j , viz., (∀OUTpr )i − (∀INpr )i − (∀DPpr )i + j (∀SCpr )i = 0. In order to update the thickness of the active layer after deposition or scour occurs, the total deposition or scour depth is compared with that of the active layer. If the total deposited layer N,M j j p=1,r=1 (DPpr )i is thicker than the active thickness (DA )i then the active layer thickness class at j + 1 is set equal to that of the newly deposited layer, viz. j +1
(DA )i
=
N,M
j
(DPpr )i .
p=1,r=1
j In case that no deposition occurs N,M p=1,r=1 (DPpr )i = 0 and only scour takes place then the updated active layer thickness is j N,M j +1 j SCpr . (DA )i = (DA )i − p=1,r=1
i
In the event that parts of the total fraction of sediments deposit and other parts of the fraction scour then the total scour depth and total deposition depth are summed up and compared to determine if net deposition or scour occurred within the cell i. The change in elevation in cell i at time j is determined as follows: N2 ,M N1 ,M j j p=1,r=1 (DPpr )i − p=1,r=1 (SCpr )i j ELi = θ where N1 + N2 = N . j
If ELi ≥ 0 then j +1
(DA )i
N1 ,M j
= (DA )i + −
p=1,r=1
j j DA i Fpr
−
N2 ,M j
(DPpr )i − j ELi θ ;
j
(SCpr )i
p=1,r=1
j
If ELi < 0 then j +1
(DA )i
N1 ,M j
= (DA )i +
p=1,r=1
N2 ,M j
(DPpr )i −
j j + Fpr ELi θ.
j
(SCpr )i
p=1,r=1
4 Model performance In this section, first, a brief description of the components of the computer code is provided; second, the performance of the
1-D hydrodynamic/sediment transport model
model for different laboratory and field data is examined; and third, a comparison of the 3ST1D model predictions with the Lopez and Falcon model is offered for field data provided by Lopez and Falcon (1999). Examination of the performance of the model is complemented with a sensitivity analysis to evaluate the importance of Manning’s resistance coefficient and the effects of cell size on the model predictions. Finally, the importance of using the total (form and grain) resistance formula in cases where well-defined step-pool configurations exist in a stream, instead of the grain resistance formula, is well demonstrated. The executable program is written in an emacs20 editor, and compiled with a NAG FORTRAN 95 compiler for Linux. The code is also tested with a FORTRAN 90 compiler on both an Alpha, and HP700 Unix machine. The initial and boundary conditions are entered to the program by means of two text files, the bed_initial.dat and the bc.dat. The bed_initial.dat file contains bed information such as, the particle sizes, and percent of each size and specific gravity in the parent and initial active layers. The program is capable of handling particles of up to five specific gravities and 20 sediment size classes. The initial conditions and boundary conditions are contained in the file bc.dat. This file contains information about the total reach length of the stream that is modeled, the stream cross sectional geometry (side slopes, bed slope, bed width), temperature, flow discharge at the upstream end, depth at the downstream end, bed porosity, number of cells, and initial value for Manning’s n. Figure 2 is a flow chart of the computer algorithm and depicts the different components of 3ST1D, the preprocessor unit where the initial and boundary conditions are provided, the main processor unit where the hydrodynamic, sediment, and update of bed changes in the active layer are processed, and the post processor unit where the output of the results is obtained. To examine the validity of 3ST1D, steep stream sediment transport data from four different sources are employed. Table 1 provides a detailed documentation of the modeled reach length L, stream (flume) gradient S, stream (flume) width W , and the median diameter d50 of the site examples. Table 1 also provides a summary of the initial and boundary conditions, namely, the flow discharge Q at the upstream end of the stream reach, the depth H at the downstream end of the stream reach, the Manning’s n, and the initial fraction Fpr of sediments found within the active layer. 4.1 Example 1: simulation of the Bathurst et al. flume study Objective: To compare the predicted and experimentally determined fraction of the material (size distribution) found atop the surface of the Bathurst et al. (1984) flume bed. The flume experiments were performed in a flume of rectangular cross-section and variable slope and is located at the Ecole Polytechnique Federale de Lausanne. The flume is 16.8 m long, 0.6 m wide, and 0.8 m high. The slope range is from −1 to 9.7 with a maximum flow capacity of 0.25 m3 /s. Table 1 provides a summary of the flow and sediment conditions of the tests presented here. Prior to the commencement of each test, material of certain size composition was placed atop the flume bed with an even
9
START
I. INITIAL CONDITIONS Read input files: bc.dat, bed_initial For each finite volume, initialize bed composition, depth, and velocity.
II. HYDRODYNAMIC COMPONENT Calculate actual depth, and velocity for every finite volume.
III. SEDIMENT TRANSPORT COMPONENT For each finite volume calculate τcr, τ∗, entrainable material, and volumetric transport capacity.
IV. BED CHANGES For each finite volume calculate scoured material, deposited material, elevation change, percent of each fraction in the active layer, characteristic grain sizes d16, d50, and d84 of the active layer, and calculate Manning’s n.
V. OUTPUT Write output to file at appropriate time (one tenth of the run time, two tenths of the run time, .... , end of run).
NO
Last Time? YES STOP
Figure 2 Flowchart of the 3ST1D model.
thickness and raked to form an initial plane bed. During the tests the volumetric sediment transport rate was recorded by measuring the volume of sediment collected in the trap at the end of each run. No sediment was fed from the upstream end of the flume into the test section; however, according to Bathurst et al. (1984) sediment was added as needed to prevent the scour holes at the headbox area of the flume from reaching the metal bed by hand from buckets. Three cases of varying hydrodynamic and geomorphic conditions are tested here with 3ST1D, namely, cases 121, 133, and 134. The reason for choosing these cases, is that Bathurst et al. (1984) performed these tests multiple times to ensure repeatability. Case 121 was conducted four different subruns (121a, 121b, 121c, 121d), case 133 for two different subruns (133a, 133b), and case 134 for four different subruns (134a, 134b, 134c, 134d). At the end of each subrun, the bed was raked prior to the commencement of the subsequent subrun. The maximum duration for all
10
Papanicolaou et al. Table 1 Summary of the available field and experimental data
Example
Bathurst et al. (1984) E-121
E-133 E-134 Cocorotico River (Lopez et al., 1999) Case 1
Case 2
Flow Q (m3 /s)
Length L (m)
Slope S (m/m)
0.08
16.8
0.05
0.06 0.07
16.8 16.8
0.07 0.07
143
1,100
0.03
143
1,100
0.03
Width W (m)
Downstream depth H (m)
Manning’s n
0.6
0.077
0.042
22
0.6 0.6
0.061 0.082
0.042 0.042
22 22
50
0.93
0.048
10
56
0.74
0.048
10 111
Kellogg Lower Creek (MacFarlane and Wohl, 2003)
1.89
48.1
0.061
2.2
0.479
0.042
Flume study Maxwell and Papanicolaou (2001)
0.139
12.2
0.03
0.762
0.152
0.042
a Specific
a Fpr
D50 (mm)
19% class with d 11% class with d 30% class with d 23% class with d 17% class with d
= 18 mm = 20 mm = 23 mm = 25.5 mm = 29 mm
0.1% class with d = 2.1 mm 49.9% class with d = 10 mm 31% class with d = 41.2 mm 10% class with d = 57.2 mm 4.5% class with d = 70.2 mm 2.2% class with d = 80 mm 0.9% class with d = 88 mm 0.4% class with d = 98 mm 1% class with d = 108.6 mm 12% class with d = 8 mm 22% class with d = 40 mm 47.5% class with d = 160 mm 18% class with d = 512 mm 0.5% class with d = 1024 mm
57.15
6.8% class with d = 5.08 mm 13% class with d = 6.35 mm 18.2% class with d = 9.65 mm 8% class with d = 25.4 mm 20% class with d = 59.8 mm 33% class with d = 67.2 mm 1% class with d = 101.6 mm
gravity is the same for all examples (= 2.65).
subruns did not exceed 110 s. The short duration of the subruns is attributed to the relatively high bedload rate that was experienced during the experiments (high gradient conditions) and lack of enough material supply. During a computer simulation run, no sediment was supplied from any source other than the bed, as there was no concern of scour reaching the bottom and no information was provided as to when or how much sediment should be added. For each run per case the flume is divided up into 70 cells and the initial and boundary conditions are supplied to the code from Table 1. In order to provide a direct comparison of the Bathurst et al. (1984) sediment transport data with those predicted by the 3ST1D model, the numerical model was run for the same durations. Table 2 offers a comparison of the recorded and predicted sediment transport data for cases 121, 133, and 134. Column 4 of Table 2 provides the calculated by the model volumetric sediment transport rate while columns 5 and 6 provide the error encountered during these runs. The average error in all cases does not exceed 7%, which is satisfactory considering the degree
Table 2 Comparison of predicted and measured data for the Bathurst et al. (1984) study (1) ID number
(2) Time (s)
(3) Measured Qs (m3 /s)
(4) Calculated Qs (m3 /s)
(5) Error
121a 121b 121c 121d 133a 133b 134a 134b 134c 134d
67 75 97 110 62 80 56 61 76 97
0.00054 0.00051 0.00054 0.00051 0.00098 0.00095 0.00101 0.00098 0.00098 0.00094
0.00051 0.0005 0.00049 0.00048 0.00092 0.00089 0.00093 0.00092 0.0009 0.00089
0.056 0.020 0.093 0.059 0.061 0.063 0.079 0.061 0.082 0.053
(6) Average error
0.0566 0.0622
0.0688
of difficulty involved in conducting the tests for high slope conditions and the fact that no sediment supply was considered during the numerical runs.
1-D hydrodynamic/sediment transport model
Figures 3(a)–(c) depict the average surface grain size distribution for cases 121, 133, and 134. These figures illustrate that the predicted final bed surface size distributions closely describe (within an error of less than 2%) those reported by Bathurst et al. (1984). The model error is well within reasonable limits given the uncertainty of the data.
4.2 Example 2: simulation of the Lopez and Falcon field study Objective: Calculation of bed changes in the Cocorotico River, Venezuela and to provide a comparison of 3ST1D and the Lopez and Falcon model. Lopez and Falcon (1999) used their 1-D model to (1) predict the response of the Cocorotico River, a small mountain stream located in the northwest region of Venezuela, to the removal of the coarsest fraction of sediment due to mining activities; and (2) illustrate the ability of their model to simulate the armoring process that is believed to occur in the Cocorotico River. The hydrodynamic component of the model is a quasi-steady and assumes subcritical flow. The sediment component of the model assumes a circular initial size distribution following the recommendation of Sardi (1973) that clasts in mountain streams follow a circular distribution. The initial size distribution is given by the following relation (Lopez, 2002, personal communication) √ 2 2d − d 2 − 2dmin − dmin Fi (d) = 2 1 − 2dmin − dmin where the ratio d = d/dmax ; Fi (d) is cumulative fraction of the sediment material with diameter d found within the cell i; dmin the minimum sediment diameter found within cell i, and dmax the maximum sediment diameter found within cell i. The Aguirre-Pe and Fuentes (1993) equation is used by Lopez and Falcon to determine the critical diameter of the bed surface material (Equation (5) in Lopez and Falcon, 1999). The data provided by Lopez and Falcon and Lopez (2002, personal communication) for the Cocorotico River include: the length of the modeled stream reach, the longitudinal gradient of the stream, the number of cross-sections within the stream reach, the width and the initial bed elevation of the cross-sections that are modeled, the maximum size distributions per cross-section, and hydrologic data that are used in their model. Some of the data are summarized in Table 1. The stream section of interest is represented by three reaches, namely, reach 1 which is the upstream reach, with length = 1200 m, reach 2 which is the middle reach, with length = 1100 m, and reach 3 which is the downstream reach with a length = 1463 m. Reach 1 has on an average about 14 cross-sections, reach 2 has about 13 cross-sections, and reach 3 has about 17 cross-sections. The average spacing between crosssections is 85 m. Reach 2 is the reach that Lopez and Falcon have simulated and the same reach is simulated here with 3ST1D. Reach 2 contains a cross-section at location x = 1475 m from the upstream end where the maximum erosion was recorded to occur and for this reason has been the focal point of both studies. In order to facilitate a comparison between the two models, the initial and boundary conditions are chosen to be identical. For
11
those parameters (e.g. gradient, channel width) that a specific value is not provided by Lopez and Falcon, an average value is assigned. In the upper half of the stream section (reach 1 and part of reach 2), the bed slope decreases from 6 to 3%, decreasing to 1% at the end of reach 3. The average representative slope, therefore, for reach 2 is assumed to be equal to 3%. Lopez and Falcon (1999) report that the bed width within the Cocorotico River ranges from 8 to 56 m and the side slopes are between 0.1 and 1.25. A careful inspection of the data provided by Lopez (2002, personal communication) indicates that the bed width (ancho) within reach 2 (cross-section 15 through 28) varies from 22 to 64 m. Due to lack of a definite width value, it is assumed here that the channel cross section is trapezoidal with an average bed width of 56 m (or 50 m). The side slopes are considered constant and equal to 0.5, and the d50 is 10 mm. Table 1 provides the initial size fraction distribution Fpr , and the flow 143 m3 /s. 3ST1D is run for 650 h, to simulate 3 years of high flows. The run is broken up by the points of major bed change as presented by Lopez and Falcon (1999). Since the stream ends in a series of small dams that were built to control sediment transport, the critical depth is considered to be the appropriate downstream boundary condition (Lopez and Falcon, 1999). 3ST1D simulations are performed for two scenarios (Table 1). The differences between scenarios 1 and 2 are found in the assumed value for the width of the channel and the choice of the Shields parameter. In scenario 1, hiding effects are considered and the Shields parameter for a sediment class pr τcr∗ pr is determined from Eqs (18) and (19); while in scenario 2, no hiding effects are considered and it is assumed that the Shields parameter for the median diameter d50 τcr∗ 50 is the representative parameter for defining sediment incipient motion for all sediment classes. Figure 4(a) and (b) presents the change in bed elevation and bed surface size composition predicted by the 3ST1D and the Lopez and Falcon model for scenario 1. The Lopez and Falcon model predicts that the bed elevation changes in the form of a step-function over the 3-year period. Instead 3ST1D predicts a higher drop in bed elevation at location x = 1475 m and that no armoring occurs in the third year. The two models predict the same scour the second year, however, 3ST1D indicates that the scour depth is increased by 30% in the third year. Figure 4(b) reveals that the range of the size distribution predicted by the 3ST1D is wider than that of the Lopez and Falcon study and indicates the presence of a wider size distribution (particularly finer particles) in the surface bed layer. Hence, one could suggest that the discrepancy shown in Fig. 4(a) and (b) between the two models is attributed to the fact that the Lopez and Falcon model does not predict incipient conditions for multifractional sediment thereby not accounting for hiding effects. Scenario 2 is basically performed to examine the validity of the above argument. In this case, the sediment component of 3ST1D assumes that the Shields parameter for the median diameter d50 τcr∗ 50 is the representative parameter for defining sediment incipient motion for all sediment classes. τcr∗ 50 = 0.065 assuming that the Shields diagram is valid. Figure 5(a) and (b) clearly shows that in scenario 2 both models predict very closely the scour depth at location x = 1475 m for the 3-year period and that the size
12
Papanicolaou et al. 1
(a)
Surface Grain Size Distribution: Case 121
0.9 0.8
Percent Finer
0.7 0.6 0.5 0.4 0.3 0.2
Initial Bed Surface Composition Final Measured Composition Predicted Final Bed Surface Composition
0.1 0 0
5
10
15
20
25
30
35
Diameter (mm) (b)
Surface Grain Size Distribution: Case 133
1 0.9 0.8
Percent Finer
0.7 0.6 0.5 0.4 0.3 0.2 Intial Bed Surface Composition Final Measured Bed Composition Predicted Final Bed Surface Compostion
0.1 0 0
5
10
15
20
25
30
35
Diameter (mm) (c)
Bed Surface Grain Size Distribution: Case 134
1 0.9 0.8
Percent Finer
0.7 0.6 0.5 0.4 0.3 0.2
Initial Bed Surface Composition Final Measured Bed Surface Composition
0.1
Final Calculated Bed SurfaceComposition
0 0
5
10
15
20
25
30
35
Diameter (mm)
Figure 3 (a) Case 121 average surface grain size distribution. (b) Case 133 average surface grain size distribution. (c) Case 134 average surface grain size distribution.
1-D hydrodynamic/sediment transport model (a)
(a) 0
0
100
200
300
400
500
600
–0.1 0
700
Time (hr)
100
200
300
400
500
–0.4 –0.6
600
700
Time (hr)
–0.3 Elevation (m)
–0.2 Elevation (m)
13
–0.5 –0.7 –0.9 –1.1 –1.3
–0.8
–1.5 Lopez
–1 Lopez
3ST1D
3ST1D
(b)
1
(b) 120
0.9 0.8 Fraction Finer
100
Percent Finer
80 60 40
Initial GSD L-F (1-yr) L-F (3-yr) 3ST1D (1-yr) 3ST1D (3-yr)
20 0 0
20
40
60
80
100
0.7 0.6 0.5 0.4
Initial
0.3
3ST1D (1-yr)
0.2
3ST1D (3-yrs) Lopez (1-yr)
0.1
Lopez (3-yrs)
0 0 120
Particle diameter (mm)
Figure 4 (a) Models predictions comparison between Lopez and Falcon (1999) model and 3ST1D model for Cocorotico River at x = 1475 m (scenario 1). (b) 3ST1D and Lopez and Falcon (1999) models predictions for evolution for Cocorotico River at x = 1475 m (scenario 1).
distributions of the bed surface predicted from the two models are almost identical. The close similarity in the predictions of the two models is not surprising considering that the Lopez and Falcon model assumes, based on the Aguirre-Pe and Fuentes equation that there is single critical diameter that describes the inception of sediment motion, which is a similar assumption with that of a constant τcr∗ 50 . 4.3 Example 3: simulation of the MacFarlane and Wohl field study Objective: Prediction of the step height and length for the Lower Kellogg Creek in the Washington Cascade range using 3ST1D. The focus of this example is to compare the field measurements regarding step spacing and step height for the Lower Kellogg stream with the 3ST1D predictions for the same stream. The Lower Kellogg is a stream with non large woody debris present and therefore, formation of step pool sequences occurs naturally. The stream is located in the Kapowsin Tree Farm in the Nisqually and Puyallup watersheds of the Washington Cascade range. Table 1 summarizes the initial and boundary conditions for the Lower Kellogg stream, which are provided by MacFarlane
0.02
0.04
0.06
0.08
0.1
Grain size (m)
Figure 5 (a) Models predictions comparison between Lopez and Falcon (1999) model and 3ST1D model for Cocorotico River at x = 1475 m (scenario 2). (b) 3ST1D and Lopez and Falcon (1999) models predictions for evolution for Cocorotico River at x = 1475 m (scenario 2).
and Wohl (2001, personal communication) and MacFarlane and Wohl (2003). The mean annual discharge for the Lower Kellogg stream is determined from the closest USGS station 12103380, which is located near Lester, WA, by computing the ratios between the drainage areas for the gaged to the ungaged station and multiplying it by the mean annual discharge obtained from the USGS station 12103380. In addition, MacFarlane and Wohl provide the values for the average step height and average step spacing, which are summarized in Table 3. The following runs are performed with the model: (1) run 1 which requires use of the semiempirical formula of Maxwell and Papanicolaou (2001) for determining total resistance, Eq. (14); and (2) run 2 which requires use of the grain resistance formula developed by Aguirre-Pe and Feuntes (1993), Eq. (15b). Both runs are compared against the measured data and they lasted about 9 h with the initial bed configuration being a flat bed. Figure 6 illustrates the variation of the bed elevation in the longitudinal direction as it is predicted by 3ST1D and measured by MacFarlane (2001). The results show that when the total resistance formula is used (Eq. 14) 3ST1D predicts very closely the average step height and spacing. In this case, the error in step height prediction is less than 4.2% and the error for the step spacing is less than 9%. When the grain resistance formula is
14
Papanicolaou et al.
Table 3 Comparison of predicted and measured data for step height and length using the MacFarlane and Wohl (2003) field data and the Maxwell and Papanicolaou (2001) flume data Studies
Number of cells
Kellogg Lower Creek (MacFarlane and Wohl, 2003) Equation (14) Equation (15b) Flume study Maxwell and Papanicolaou (2001)
Running time (h)
Measured
Predicted
Error (%)
d¯step (m)
L¯ step (m)
d¯step (m)
L¯ step (m)
d¯step error
L¯ step error
8.3 70
200 200
9 9
0.20 0.20
2.4 2.4
0.208 0.214
2.6 4.1
4.2 6.9
50
9
0.048
1.98
0.055
2.16
14.8
9.09
3.5 Measured Bed 3 Maxwell and Papa.
Elevation (m)
2.5
Agruirre-Pe and Fuentes
2
1.5 1 0.5 0 0
10
20
30
40
50
60
Logitudinal distance (m)
Figure 6 3ST1D model predictions for bed morphology for Kellogg Lower Creek using Maxwell and Papanicolaou roughness formula and Aguirre-Pe and Fuentes roughness formula.
used, the step height is overpredicted by almost 7% and the step spacing by 70%. The grain resistance formula underpredicts the friction factor f and therefore the model overpredicts the spacing of steps. According to Millar (1999) up to 90% of f at bankfull flow can be attributed to form roughness in the presence of step pool configurations. 3ST1D is tested against other field data (e.g. Pioneer Creek in the Green river watershed, WA) provided by MacFarlane (2001) but due to space limitations are not included here.
4.4 Example 4: simulation of the Maxwell and Papanicolaou laboratory study Objective: To perform a sensitivity analysis of the model. The Maxwell and Papanicolaou (M-P) (2001) experimental study is employed here to test the sensitivity of 3ST1D using (1) different cell numbers; and (2) different Manning’s roughness coefficient values. The M-P experimental study focuses on the evolution of step pool formation and offers detailed information about the initial bed surface size composition, the median diameter, the flow discharge, the average depth, the flume gradient
(see Table 1) and the step pool characteristics (step height and spacing). Figure 7 demonstrates the model sensitivity analysis to cell size using the M-P (2001) flume data. Comparison of model runs using different cell sizes ranging between 50 to 150 cells indicates that 3ST1D model is in general insensitive to the choice of number of cells as long as ti > t. For example, the choice of 50 and 100 cells in Fig. 7 yields bed profiles that have an average value for the root mean square error (RMSE) equal to 4.8 and 5.0%, respectively. In short, as the element size becomes very small, model predictions become insensitive to this parameter since the model stability criterion is satisfied (Eq. 13). Sensitivity analysis for 3ST1D model predictions to the roughness coefficient is also investigated. For the sake of an example, Manning’s n values of 0.042 and 0.084 are used to investigate the sensitivity of the results to a 100% variation in the parameter value. Modeling results show that bed profiles are sensitive to the choice of Manning’s n (Fig. 8) and that the model responds differently but appropriately for higher Manning’s n values. The bed profile that is estimated with Manning’s n = 0.042 (RMSE = 4.8%) represents closer the step-pool patterns recorded during the tests than that estimated with n = 0.084
1-D hydrodynamic/sediment transport model
15
100.4 RMSE(50 cell)= 4.8% 100.35
RMSE(100 cell)=5.0%
100.3
Bed Elevation (m)
100.25
100.2
100.15
100.1
100.05
100
99.95 0
2
4
6
8
10
12
14
Longitudinal distance (m) 50 cell
100 cell
Final Measured Bed
Figure 7 3ST1D model sensitivity analysis to cell size using the Maxwell and Papanicolaou (2001) flume data. 100.4 bed_n=0.084
100.35
bed_n=0.042
100.3
Final Bed
Elevation (m)
100.25
Initial bed
100.2
RMSE (n = 0.042) = 4.8%
100.15
RMSE (n = 0.084) = 7.4% 100.1 100.05 100 99.95 0
2
4
6
8
10
12
14
Longitudinal distance (m)
Figure 8 3ST1D model sensitivity analysis to Manning’s n value using the Maxwell and Papanicolaou (2001) flume data.
(RMSE = 7.4%). In the latter case, the bed profile is almost flat due to the presence of high bed roughness. As expected high roughness coefficient yields to a higher flow depth and consequently higher erosion. 5 Conclusions A one-dimensional numerical model is developed for calculating flow and sediment transport in steep mountain streams. 3ST1D consists of two coupled components, the hydrodynamic and
the sediment transport. The hydrodynamic component assumes unsteady flow by solving the unsteady form of the SaintVenant equations. The Total Variation Diminishing Dissipation (TVD)-MacCormack scheme is employed here to approximate the hydrodynamic solution over transcritical flow stream reaches. The sediment component of the model accounts for multifractional sediment transport and sediment entrainability. Multifractional sediment characteristics are accounted for by using the Komar (1989) method and the Graf and Suszka (1987), which identifies sediment incipient conditions for low relative
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submergence, a condition typically encountered in mountain streams. The capacity of sediment is predicted by the Schoklitsch method while sediment entrainability is accounted for based on a state-of-the art formula that accounts for the bed porosity, bursting frequency, probability of occurrence of highly episodic turbulent events, the area of bed section, and sediment availability in the unit bed area. The overall performance of the model is evaluated by comparing its predictions with observations from two flume studies, two field investigations and against the predictions of the quasi-steady model of Lopez and Falcon developed for mountain streams. The model favorably predicts the grain size distribution of the experimental data of Bathurst et al. (1987) study within an error of less than 2%. A direct comparison against the Lopez and Falcon model predicitions shows that 3ST1D accuretly predicts incipient conditions for multifractional sediment thereby accounting for hiding effects. On the contrary, the Lopez and Falcon model assumes, based on theAguirre-Pe and Fuentes equation, that there is single critical diameter that describes the inception of sediment motion. 3ST1D predicts very closely the average step height and spacing provided by the MacFarlane and Wohl (2003) field data. 3ST1D gives accurate predictions when the total roughness formula is used, however, if the grain resistance formula is used, the step height is overpredicted by almost 7% and the step spacing by 70%. The sensitivity of the model is investigated by using the Maxwell and Papanicolaou (2001) flume data. The analysis is performed to assess the effects of cell size and Manning’s roughness coefficient value in the predictive ability of the model. Results showed that 3ST1D is overall insensitive to the choice of cell size when ti > t. Moreover, for different Manning’s roughness coefficient values the model responds appropriately. A higher value of Manning’s roughness coefficient results to a flattened bed comparatively to a small value. Although more testing and calibration of the model may be necessary in order to improve its predictive ability, it is expected that the coupled hydrodynamic/sediment transport 3ST1D model would become a valuable predictive tool to watershed managers, aquatic engineers, and those who are interested in predicting sediment routing and bed deformation in mountain streams. It is expected that this model can be used as a preliminary design tool for river restoration projects and can facilitate quick computer runs for long stream reaches, particularly when long-term simulations of long-term effects are needed in a relatively short period. Future research will include the conversion of the model in 2-D flows to remove some of the limitations of the current modeling work. Acknowledgments The authors of this paper are grateful to Dr Ellen Wohl, Mr William MacFarlane, Dr Jose L. Lopez, Dr Bathurst and Mr Adam Maxwell for graciously providing their data for use in this study. This investigation was partially funded by the PNNL hydrologic group (Dr Richmond) and by the Nez Perce Indian Tribe, ID The comments by Dr Wohl are highly appreciated and have considerably improved the quality of the manuscript.
Notation A[x, H (x, t)] = wetted cross-section area AB = the average area of the bursts Abed = stream bed area AN = the normalized area of a burst k a¯ i+1/2 = discrete average characteristic velocity of the states at i and i + 1 B = width of the cell c¯i+1/2 = discrete approximations to the wave celerity Cn = the Courant number dstep = the average step height d16 = the size of sediment for which 16% of the sample is finer d84 = the size of sediment for which 84% of the sample is finer d50 = the size of sediment calculated in inches for which 50% of the sample is finer d¯pr = the median particle diameter of a fraction with size p and density r dmax = maximum sediment size exists in the bed material DA = the active layer thickness DPpr = deposition depth E = the rate of sediment erosion Fpr = the fraction of particles with size dpr present in the active layer Fi (d) = cumulative fraction of the sediment material with diameter d found within the cell i f = total resistance f = grain resistance f = form resistance g = acceleration of gravity H (x, t) = water depth j H˜ i = the value of depth at the end of the predictor part ≈j
H i = the corrected value of depth at the end of the corrector step H /d50 = the relative submergence i = denotes space j = denotes time L = the average step length m = hiding function exponent M = maximum number of sediment density classes present in the mixture n = Manning’s roughness coefficient N = maximum number of sediment size classes present in the mixture PE = the probability of entrainment Q(x, t) = discharge qsb = volumetric bedload transport rate per unit width q = flow discharge per unit width qc = critical discharge
1-D hydrodynamic/sediment transport model j
Ri = the hydraulic radius SCpr = scour depth S0 = bottom gradient Sf = friction gradient TB = average bursting period (TFRAC )i = time fraction t/ti ti = the travel time τcr∗ 50 = dimensionless critical shear stress of the d50 sized-sediment τcr∗ pr = dimensionless critical shear stress for a fraction with size p and density r τN = the normalized instantaneous stress τ = time-averaged component of the bed shear stress τ = fluctuating component of the instantaneous effective bed shear stress U∗ = the shear velocity j U˜ i = the value of velocity at the end of the predictor part ≈j
U i = the corrected value velocity at the end of the corrector step u¯ i+1/2 = discrete approximations to the local water velocity ρs = density of the sediment ρ = fluid density ν = kinematic viscosity θ = porosity of the stream bed j (∀ACTpr )i = the volume of the available material within the active layer thickness DA of the cell i ∀INpr = the volume of sediment particles with size class p and density r entering the cell i j (∀FRACTION_TOTAL )i = the total volume for all fractions available in the cell bed j (∀SCpr )i = volume of scour within the cell i j (∀OUTpr )i = the volume of sediment with class p, r that exits within the cell i j (∀DPpr )i = the volume of deposition within the cell i j ELi = changes in bed elevation within cell i t = the time step between two subsequent nodes x = the cell space between two subsequent nodes References 1. Abt, S.R. et al. (1987). “Development of Riprap Design Criteria by Riprap Testing in Flumes: Phase I”. NYREG/ CR-4651. US Nuclear Regulatory Commission, Washington, DC. 2. Aguirre-Pe, J. and Fuentes, R. (1993). “Stability and Weak Motion of Riprap at a Channel Bed”. In: Thorne, C., Abt, S., Barends, F., Maynord, S. and Pilarczyk, K. (eds), River, Coastal and Shoreline Protection. Erosion Control using Riprap and Armourstone. Wiley, New York, pp. 77–92.
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