Sediment transport models




Introduction

The hydrological, geomorphological, environmental and ecological state of streams and rivers occur over a range of spatial and temporal scales and are good indicators of the health of the system. Healthy riparian and animal communities depend on the change in flows, shifting channels, and moving sediments to provide inputs of organic and mineral materials. These same drivers also are key to physically shaping the stream or river system; they are what form, maintain, and alter the channels. All surface water resource projects impose some changes on this dynamic system whether it be water velocity and depth; the concentration and size of sediment particles moving with the water; or the width, depth, slope, hydraulic roughness, planform and lateral movement of the stream channel (USAC, 1989). Understanding this is an important, but sometimes overlooked, part of stream restoration, which is where sediment transport modeling comes in.

Basics

LanesBalance.jpg
Figure 1: Lane's Balance




Channels are formed, maintained, and altered by two things; flows and sediment loads. Equilibrium is achieved through a balance of four factors, as shown in Equation 1:

sediment discharge, sediment particle size, streamflow, and stream slope (Lane, 1954).

Lane.jpg
Equation 1

Where

Lane_variables.jpg
This equation qualitatively states that the sediment load, which is the first half of the equation, is proportional to the stream power which is represented by the second half of the equation. Equilibrium occurs when the streamflow power is constant over the length of the stream resulting in zero change in the shape. By changing any term on either side of the equation, the balance is shifted and one or more of the other variables must compensate for this, as shown in Figure 1. Reaching equilibrium usually involves erosion (Loucks, 2005). An example is a stream below a dam - the effluent from the dam is going to be sediment starved so the intial Qs is low. Qw, the streamflow, can't be naturally adjusted so equilibrium is reached through changes in channel slope, S, the mean sediment particle size, D50, and picking up sediment from the channel bed immediately below the spilway. Bed armoring can result.
Types_of_Sediment_loads.jpg
Figure 2 - Types of sediment loads in rivers (Louck, 2005)


An equilibriating stream tends to erode more sediment and larger particle sizes, resulting in erosion and downcutting in some areas and aggredation in others. The channel evolution model helps explain this. The transported sediment can be dissolved, suspended and pushed along by saltation and traction. The suspended load is usually the fine particles that make a stream look muddy, like silt and clay, and can make up as much as 95% of the sediment carried by the stream (Louck, 2005). Saltation and traction are the two processes that form the bedload (see Figure 2, Louck, 2005). For saltation and sliding or traction to occur the flow must reach a critical velocity that is dependent on the particle size and material. This corresponds to shear stress and is fundamental in understanding sediment transport and the modelling process.

Bed Shear Stress


For a particle to become entrained, the bed or boundary shear stress caused by the water flowing parallel to the stream bed must overcome a critical shear stress. It can be thought of as a force balance - if the applied force of the water (primarily hydrodynamic drag, FD but also hydrodynamic lift, FL) overcomes the resistive force of the submerged weight of the particle, FG, the particle will become entrained, as shown in Figure 3. The threshold when the two forces are equivalent is the critical condition at which the applied forces are just balancing the resisting forces (Chang, 1988).


SchematicSaltation.jpg
Figure 3: Forces acting on a particle (van Rijn, 1984)

The following is a breakdown of the different forces:
hydrodragF.jpg
Equation 2a

Where c1 is a constant found experimentally, τ0 is the bed shear stress or tractive force, and d is the grain diameter. The effective surface area that the shear stress is exerted upon is equal to c1τ0d2. This force acts at the center of gravity of the particle.

submergedWt.jpg
Equation 2b

Here c2 is also a constant found experimentally, γs is the specific weight of the sediment, and γ is the specific weight of water. If the bed of the channel is sloped, the angle formed with the horizontal is designated as Φ and θ is the angle of repose or friction angle between the submerged particles. Right before the particle starts to move the resultant of these forces FD and FG is in the direction of the friction angle so the ratio of forces acting parallel to the bed versus those normal to the bed is equal to tan θ. This simplifies to Equation 2c:
tantheta_sed.jpg
Equation 2c

Combining Equations 2a, 2b, and 2c results in the critical shear stress τc, Equation 3:
CriticalShearStress.jpg
Equation 3

And for a horizontal bed:
horizontalBed.jpg
Equation 4

ShieldsCurve.jpg
Figure 4: Shield's Diagram (Cao, 2006)
where the left hand side represents the ratio of the hydrodynamic force versus the submerged weight (Chang, 1988).

Shield's Diagram


Dimensional analysis on a particle leads to two dimensionless numbers; the Shield's stress ,τ*, which is essentially Equation 4 is re-written as
ShieldsNumber.jpg
Equation 5

and a dimensionless viscosity, or Reynolds number Re* that is a function of the particle diameter and density.
Reynolds_dim.jpg
Shield obtained a relationship between the dimensionless critical stress and the dimensionless viscosity using experimental data. This is known as the Shield's Diagram (see Figure 4). The variation of τ* with Re*demonstrates the effect of fluid viscosity on grain movement. Since τc can't be measured directly, this empirical method gives criterion for incipient motion while only needing to solve for a specific version of the Reynolds number, usually through trial and error with a known set of sediment and fluid values. The Shield's diagram holds for sand and gravel and non-cohesive particles, but is not as effective for clays and fines that clump together because fines are more poorly sorted, have electrostatic forces, and are regulated more by turbulent movement (Chang, 1988 and Wilcock, 2004). The Bureau of Reclamation has a great manual with more information on the differences between cohesive and non-cohesive sediment transport here.

Sediment Transport Capacity

GrainsizeBedLoad.jpg
Figure 5: Grain sizes associated with bed load, bed-material load, suspended load, and wash load. (Wilcock, 2009)




Transport capacity is the rate at which the stream or river moves sediment at a given flow. As mentioned previously, the two main mechanisms of sediment transport are: 1) bed load, where the grains move along the bed by sliding, rolling, or hopping; and 2) suspended load, where grain are picked up off the bed and move along a more turbulent path. In many streams, grains smaller than 1/8 mm are always suspended while grains great than 8 mm travel as bed load. The strength of flow determines the transport mechanism of grains in between these two sizes. Sediment transport can also be categorized based on the source of the grains: 1) bed material load, which is grains found in the stream bed; and 2) wash load, which is finer grains found as less than a percent or two of the total amount in the bed (Wilcock, 2009). Figure 5 provides a good visual of the different grain sizes associated with each transport mechanism. For this paper, bed load and sediment load will be the main transport mechanisms considered. It is also important to keep in mind that the boundary between the two is not absolute; it really depends on the flow strength. Formulas used to describe the two for steady uniform flow have been developed based on field calibrations and flume data. The most commonly used follow.

Bedload Formulas


Several bedload equations and their assumptions follow.

TABLE 1
Equation Name
Concept
Shear Formula
Bed-load Discharge
Variables
Assumptions
DuBoys Formula
Relates bed-load discharge per unit channel width,qb,to the excess shear stress, or τ0- τc.
DuBuysShear.jpg
DuBuyDischg.jpg
DuBuyVar.jpg
Cd and τc were obtained upon experiments in small lab flumes by Straub.
  1. Uniform sediment grains move as superimporsed layers with a thickness
  2. At the threshold for incipient motion n=1
Shields Formula
Also based on the excess shear stress.
ShieldsDimen.jpg


  1. The left hand side of this equation represents the dimensionlessbed-load discharge while the right hand side lumps the excess shear stress and the submerged weight of the sediment particle.
**Table contents modified from Chang, 1988

Suspended Load Formulas


Einstein developed the method most commonly used to evaluate the suspended load. For more information, see the Sediment Transport page. It is important to have a general understanding of the different equations (and there are many!) and what assumptions are made for each as well as what conditions they are best suited for when doing sediment transport modelling. Most modelling programs have different transport methods that can be chosen based on whether the channel is sand or gravel, the grain size distribution, or how well sorted the bed is.

Numerical Models




Shields Diagram gives an empirical way of approximating the sediment load which is good for getting an estimate or basic understanding of the system with limited data. For a more in-depth understanding, numerical models are used with increasing frequency. The selection and application of the model is strongly dependent on the type and scale of the problem being studied. There are initial or sediment transport models that compute the sediment transport rate and bed level changes for one time step, resulting in a short-term prediction, and there are dynamic morphological models that compute the flow velocities, wave heights, sediment transport rates, bed level changes and velocities as a continuous loop. There are models that look specifically at bed deformation, some look at channel evolution/bank stability, and then there are others that combine the two. There are also one, two and three dimensional simulation methods (van Rijn, 1984).

1-D Models


One dimensional models are commonly used in situations where the flow field shows little variation over the cross-section, like flow in some river systems. This is the most commonly used method for sediment transport studies.

CCHE1D is one of the commonly used, free 1-D transport models available. One really nice thing about CCHE1D is that it can be integrated with GIS to process topographic data and generate model input data. The governing equations are St. Venants 1-D equation (Equations 6 and 7), a sediment continuity equation (Equation 8), and a channel bed deformation equation (Equation 9) as shown below.
CCHE1D_Eqns.jpg
Equations 6, 7, 8, and 9
nodal_config.jpg
Figure 6: Computational Cell (Wu, 2002)

ExtractedChan_CCHE1D.jpg
Figure 7: Extracted Channel Network, Goodwin Creek (Wu, 2002)

CCHE1D_Nodes.jpg
Figure 8: Refined Computational Grid for Goodwin Creek (Wu, 2002)




For these equations, x and t are the spatial and temporal axes; A is the flow area; Q is the flow discharge; h is the flow depth; So is the bed slope; ß is a correction coefficient for the momentum due to the nonuniformty distribution at the cross section; g is the gravitational acceleration; and q is the side discharge per unit channel length. To actually model this, the continuity equation and the momentum equation (Equation 7) are both discretized, or broken from a continuous equation into a series of discrete "nodes" as shown in Figure 6, using the Preissmann four-point scheme and then solved iteratively. Basically, it takes input as shown in Figure 7 and then assigns nodes along the stream, To solve these conditions, upstream boundary conditions at the inlets and downstream conditions at the outlet of the channel network as well as internal conditions at confluences and hydraulic structure locations are necessary. The inflow boundary is defined by either a hypothetical hydrograph or a given time series of discharge. For the outflow, a stage-discharge curve or a time series of stage is imposed. Flows through structures are complicated and difficult to simulate using 1-D models so simplifications are made. An example of output results are shown below in Figure 9 (Stone et al., 2007 and Wu, 2002).
CCHE1D_Results.jpg
Figure 9: Measured and computed annual sediment yield for Goodwin Creek (Stone, 2007)

HEC-RAS (SIAM) also uses St. Venant equations and the Preissmann four-point scheme to describe flow. Transport potential is computed by grain size fraction which allows simulation of non-uniform sediment movement and bed material size change. This is especially applicable to get an understanding of the long-term effects of scour and deposition. It is best suited to steady, equilibrium sediment transport. This model works well for making a sediment budget analysis (Stone et al., 2007).

SAM is another commonly used tool developed for the Army Corps of Engineers available through Owen Ayers & Associates, Inc. The main purpose of SAM is to calculate stable channel dimensions that will pass a prescribed sediment load without deposition or erosion. It uses a package of three different design modules, SAM.hyd; SAM.sed; and SAM.yld that each build on one another, starting with SAM.hyd. SAM.hyd can solve for any the the variables in the uniform flow equation depending on what the user specifies as the dependent variable. The default channel method (alpha) assumes steep banks are vertical and have no influence so calculations are performed for the channel bed only which can cause a huge variation in results for channels that are narrow and steep. SAM.sed uses hydraulic input that is either from SAM.hyd or user specified along with bed gradation to calculate a sediment discharge rating curve. The sediment transport function is applied at a point which does not allow for variability in sediment distribution with time or space so it's possible that the calculated transport rates are inaccurate. It is important to choose the proper sediment transport equations based on the bed gradation (i.e, gravel vs. sand, well sorted vs. poorly). SAM.aid, a module for use with the SAM package, is helpful in determining what transport function to use based on the stream conditions. SAM.aid is especially useful for low-budget projects because it can be used with limited field data. SAM.yld calculates the sediment that passes through the cross section for some time period, be it a flood event/single storm or an entire year. The sediment discharge rating curve created by SAM.sed is used in conjunction with the flow duration curve or hydrograph to get a representative value of sediment discharge (Thomas, 2002).

The main difference between SAM and CCHED-1 is that it represents the system as an average. This simplification makes it easy and quick to use, but limits its usefulness. It is mostly used as a tool during planning that can help determine the slope, channel design, rip-rap etc. for a stable channel.

Other available 1-D models include Mike 11, Fluvial12, and GSTARS3.


2-D Models


Two dimensional models can be either depth averaged (2DH) or vertically averaged (2DV). Depth averaged simulations are useful when the flow field has no significant variations in vertical direction and where the fluid densities are constant. For stream modelling these are useful because they allow properties associated with non-uniform, meandering flow and flow near hydraulic structures to be incorporated. There is also no need for a momentum correction coefficient, unlike most 1-D models. 2-D flow in the vertical plane is useful when the flow is uniform in one lateral direction but has significant variations in the vertical direction, such as flow accross trenches or long crested dunes (van Rijn, 1993; and Stone et al., 2007).

CCHE2D uses the depth-integrated, two-dimensional flow momentum equation for turbulent flow in Cartesian coordinates.
CCHE2D_GovEqn.jpg
Equations 10, 11, and 12

Equation 10 is the depth integrated continuity equation which is used to calculate the free surface elevation for the flow. For Equations 10-12, U and V are the depth integrated velocity components in the x and y directions, respectively; t is the time; g is gravitational acceleration; h is the local water depth; ρ is water density; fcor is the Coriolis parameter and τxx, τxy, τyx, and τyy are the depth integrated Reynolds stresses; τbx and τby are shear stresses on the bed surface. This is represented by Equation 13, shown below.

CCH2D_Coriolis.jpg
Equation 13

Here vt is the eddy viscosity coefficient. The stresses are approximated using the assumption that they are related to the main rate of strain of the depth-averaged flow field with a coefficient of eddy viscosity.
CCH2D_SedTrans.jpg
Equations 14, 15, and 16

Equations 14-16 are used for sediment transport processes, where Ck is the suspended sediment concentration in the kth size fraction and εs is the turbulence diffusivity coefficient. These equations are solved by discretization, as explained for 1-D models. The method used by CCHE2D is called the Efficient Element Method.

Examples of output from CCHE2D are shown below in Figure 10 and Figure 11.


2-D_BedChgSim.jpg
Figure 10: CCHE2D simulation of bed change in meandering channel (Stone et al., 2007)

2-D_FlowField.jpg
Figure 11: CCHE2D Simulation of flow field in East Fork River (Stone et al. 2007)

Other available 2-D models are Mike 21C and SED2D.

3-D Models


Three dimensional models are of particular interest when there is a lot of variation in the vertical direction; structures in a channel are a good example of this. The most general hydrodynamic model to describe the flow in a specific control volume is a three dimensional, time-dependent model. The different processes can be described in terms of balances, ie mass balance, momentum balance, etc. Other aspects of the fluid's behavior can be described using empirical equations such as those connecting the fluid density to the temperature and salinity of the fluid. The effects of small-scale turbulent motions on the time-averaged flow are also represented by empirical equations connecting shear stresses to velocity gradients (eddy viscosity) (van Rijn, 1993).

CCHE3D solves the Navier-Stokes equation.

cche3d_navierstokes.jpg
Equations 17 & 18

Equation 18 includes the velocity components, ui; Fi, the gravity force per volume; the fluid density ρ; and the pressure p. Turbulent stresses τ are calculated using the turbulent kinetic energy and its dissipation rate. The sediment model equations include 3D sediment advection-diffusion and bed deformation. The equations are discretized and solved using the Efficient Element Method. Like many other 3D models, it is especially useful for determining scour around hydraulic structures and sediment transport in areas that have strong spatial variability (can't assume flow in one direction is negligible). Figures 11 and 12 show examples CCHE3D simulation of scour around cylinders (Stone et al., 2007).

cche3d_flowfield.jpg
Figure 12: Flow field and scour around cylinder modeled with CCHE3D (Stone et al., 2007)
cche3d_topography.jpg
Figure 13: 3D topography of scour hole using CCHE3ED (Stone et al., 2007).


Delft3D is another 3D modelling suite that can be used to model flow, sediment transport and morphology, waves, water quality and ecology as well as modelling their interactions. It is used mostly for modelling coastal and estuarine areas where it can be used to understand storm surges, density driven flows, salt intrusions, transport of dissolved material and pollutants, and sediment transport and morphology, among other things. The ability to model water quality, including the adsorption and desorption of contaminants and the deposition and suspension of adsorbed substances to and from the bed add a novel and more complicated element to 3D modelling, which is already complicated to begin with. A technical manual is not available for the software since licences are only available for purchase.

3D sediment transport modelling is still in its infancy and is not common because of how expensive it is both for data collection and computation. Other available 3-D models are EFDC and Mike 3.

Summary of Numerical Models


Dimensionality
Governing Equations
Assumptions
Pros
Cons
Best Use
Available Software
1-D
St Venant Equations
Average the 3-D equation over the cross-section so only longitudinal flow is simulated
  1. Less computationally expensive
  2. Data collection also less expensive
  1. Doesn't give the big picture
Flow in rivers where there is little variation over the cross section; narrow but shallow streams.
  1. Mike 11
  2. Fluvial12
  3. GSTARS3.
  4. CCHE1D
  5. HEC-RAS (SIAM)
2-D
2 Dimensional Momentum Equation
Average the 3-D governing Equations in the vertical or transverse direction (ie depth is much smaller scale than the reach, laterally and longitudinally)
  1. Not as computationally expensive as 3-D models
  1. Some accuracy sacrificed
  2. Still requires a lot of data, which can be cost prohibitive
Flow in channels where planform variation is important or flow is unsteady.
  1. Mike 21C
  2. SED2D
3-D
3 Dimensional Momentum Equation or
Navier-Stokes Equation

  1. Most accurate
  1. Involve intensive computation, require a lot of data
Local scour problems, lake, estuary, and reservoir environments
  1. EFDC
  2. Mike 3
  3. Delft3D

When selecting a model to use, it is important to have a clear idea of what the goal of the model is and what data is available. Quality of data is also important; low quality data will do nothing to improve the model, rather it might cause unforeseen errors. This helps determine whether the model should be 1-D, 2-D, or 3-D. It is also helpful in determining what software to use. Duan et al. (2008) prepared an evaluation of the Rillito River for the Pima County Regional Flood Control District using four 1-D models (ILLUVIAL 2, HEC-RAS 4.0, HEC-6, and SRH-1D) and compared the computational data to observed. They found that IALLUVIAL 2 produced the stage hydrograph with the smallest root mean square error (RMSE), however HEC-6 and HEC-RAS 4.0 had more accurate averaged bed elevation changes.

Applications/Case Study - Why is Sediment Transport Modelling So Important?



Elwha Dam Removal


Why is sediment transport modelling so important? We know that healthy riparian and animal communities depend on the dynamics of the channel to provide nutrients and it is these same driving forces - the flows, shifting channels, and moving sediments- that physically shape a river or stream system. Understand this interaction before starting a restoration project can help the designer to optimize the design for a desired outcome and to understand what results might be expected.

The Elwha Dam removal is a classic example of this. Before the dam was removed, Draut (2010) studied the existing system and the changes in channel evolution caused by the dam by looking at the geology and hydrology of the system both upstream and downstream of the dam over time. Below the dam the stream was incised, narrowed, and there was bed armoring. Since removing the dam releases all the trapped sediment, aggradation and bar formation on the lower Elwha can be expected. This is significant in several ways. The formation of bars could in the long term improve salmon habitat, but it could raise the 100 year flood stage more than a meter (Draut, 2010). Konrad (2009) found that while the sediment released initially from dam removal in the short term will decrease the the salmon habitat, the long term effect will be positive.

This project was successful because of the thorough understanding of the system. It included:
  1. Project Planning
  2. Site Analysis (This is demonstrated by both papers - there was a lot of effort to understand the system before the removal. The dam was removed in 2011 and these papers were published in 2010 and 2009, respectively.)
  3. Selection of Design Procedure (Use of notched removal?)
  4. Implementation
  5. Monitoring (This is occurring now)

There is a nice collection of papers on the Elwha Dam removal from NOAA that looks at both before and after here. Dams and Impoundments goes into more detail, giving a clear definition and outlining the environmental effects.


References




Delft3D-FLOW: Simulation of mulit-dimensional hydrodynamic flows and transport phenomena, including sediments. October 2007. Deltares Systems.

Engineering and design: Sedimentation and investigations of rivers and reservoirs. 1989. Washington, D.C.: U.S. Army Corps of Engineers.

Stream Restoration and Design (National Engineering Handbook 654). 2008. United States Department of Agriculture: Natural Resource Conservation Service.

Cao Z., Pender G., Meng J. 2006. Explicit formulation of the shields diagram for incipient motion of sediment. Journal of Hydraulic Engineering 132(10).

Chang H.H. 1992. Fluvial processes in river engineering. Malabar, Florida: Krieger Publishing Company.

Draut A. E., Logan J.B., Mastin M. C. 2010. Channel evolution on the dammed Elwha River, Washington, USA. Geomorphology 127(2010).

Duan J.G., Acharya A., Yaeger M., Zhang S., Salguero M. July 5, 2008. Evaluation of flow and sediment models for the rillito river.

Jia Y., Wang S.Y. CCHE2D: Two-dimensional Hydrodynamic and Sediment Transport Model for Unsteady Open Channel Flows Over Loose Bed

Konrad C.P. 2009. Simulating the recovery of suspended sediment transport and river-bed stability in response to dam removal on the Elwha River, Washington. Ecological Engineering 35(2009).

Lane E.W. December 1954. The importance of fluvial morphology in hydraulic engineering. Denver, Colorado: US. Department of the Interior Bureau of Reclemation.

Loucks D.P., van Beek E., Stedinger J.R., Dijkman JPM, Villars M.T. 2005. Appendix A: Natural system processes and interactions. In: Water resources systems planning and management. Italy: United Nations Educational, Scientific and Cultural Organization.

Stone M., Chen L., Scott S. September 2007. Guidance for Modeling of Sedimentation in Stream Restoration Projects. Desert Research Institute, Nevada System of Higher Education. Prepared for U.S. Army Corps of Engineers, Engineer Research and Development Center.

Thomas W., Copeland R., McComas D. September 2002. SAM Hydraulic Package for Channels. Vicksburg, Mississippi: Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development Center.

van Rijn L.C. 1984. Sediment Transport, Part I: Bed Load Transport. Journal of Hydraulic Engineering 110(10).

van Rijn L. C. 1993. Principles of Sediment Transport in Rivers, Estuaries and Coastal Seas. Amsterdam: Aqua Publications.

Wilcock P. Sediment Transport in Gravel-Bed Rivers with implications for channel change. Lecture notes, guest lecture at Univ. California Berkeley January 26-28, 2004.

Wilcock P., Pitlick J., Cui Y. 2009. Sediment transport primer: Estimating bed-material transport in gravel-bed rivers. Rocky Mountain Research Station: U.S. Department of Agriculture Available from: General Technical Report RMRS-GTR-22.

Wu W. January 2002. One-dimensional channel network model; CCHE1D – Technical Manual. National Center for Computational Hydroscience and Engineering. Technical Report No. NCCHE-TR-2002-1




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by A. Symonds