Modeling the Evolution of Incised Streams: I. Model ... · Modeling the Evolution of Incised Streams: I. Model Formulation and Validation of Flow and Streambed Evolution Components

Modeling the Evolution of Incised Streams:I. Model Formulation and Validation of Flow and

Streambed Evolution ComponentsEddy J. Langendoen, M.ASCE1; and Carlos V. Alonso, M.ASCE2

Abstract: A robust computational model for simulating the long-term evolution of incised and restored or rehabilitated stream corridorsis presented. The physically based model simulates the three main processes that shape incised streams: hydraulics, sediment transport,and streambed and bank adjustments. A generalized implicit Preissmann scheme is used for the spatial and temporal discretization of theflow governing equations to accommodate large time steps and cross sections spaced at irregular intervals. The solution method introducesseveral enhancements that increase its robustness, specifically to simulate flashy flows. Transport of cohesive or cohesionless graded bedmaterial is based on a total-load concept, and suspended and bed load transport modes are accounted for through nonequilibrium effects.The model simulates channel width adjustment by hydraulic erosion and gravitational mass failure of heterogeneous bank material. Thepresent paper focuses mainly on the treatment of streamflow hydraulics and evolution of graded streambeds, and reports simulations ofpublished experiments on degrading and aggrading channels with graded bed material. Description and validation of the model’s stream-bank erosion component and the application of the model to incised stream systems are presented elsewhere.

DOI: 10.1061/�ASCE�0733-9429�2008�134:6�749�

CE Database subject headings: Computer models; Channel morphology; Open channel flow; Bank erosion; Sediment transport;Streams.

Introduction

Channelization-induced stream incision is widespread in the mid-south and midwestern United States. The highly erodible soil isunable to halt the incision and subsequent widening of thesechannelized streams, leading to increased sediment productionand yields as material is eroded from the bed and banks. In north-ern Mississippi incision has particularly been severe for thosestreams draining westwards from the Loess Hills into the YazooRiver Basin. Simon et al. �2004� show that the Mississippi ValleyLoess Plains ecoregion experiences the second highest suspendedsediment yields in the continental United States. Many of thesestreams have experienced severe instability since European settle-ment in the 1830s, leading to disruption of the fluvial system andsevere impacts on local communities, farmers, and ecologicalhabitats �Watson et al. 1997�. The Federal Interagency Demon-stration Erosion Control �DEC� project was established in 1984and charged with planning, design, and construction of structuralrehabilitation measures to stabilize the channels, restore habitat,

1Research Hydraulic Engineer, Agricultural Research Service,National Sedimentation Laboratory, U.S. Dept. of Agriculture, Oxford,MS 38655. E-mail: [email protected]

2Research Hydraulic Engineer, Agricultural Research Service,National Sedimentation Laboratory, U.S. Dept. of Agriculture, Oxford,MS 38655

Note. Discussion open until November 1, 2008. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on January 12, 2006; approved on September 14, 2007.This paper is part of the Journal of Hydraulic Engineering, Vol. 134,
No. 6, June 1, 2008. ©ASCE, ISSN 0733-9429/2008/6-749–762/$25.00.
JO

and reduce sediment yield from the Yazoo River Basin �Cooperet al. 1996�. The DEC approach is largely based on geomorpho-logical concepts that depends on characterization of the rate andfrequency at which water flows through a stream channel �Dunneand Leopold 1978; Carling 1988�. Determination of how much,how fast, how deep, and how often water flows is a critical step inpredicting stream channel evolution and developing restorationinitiatives �FISRWG 1998�. Similarly, accurate evaluation of howoften and how much sediment is transported by a stream as afunction of water discharge is an important step in establishing ascientifically defensible strategy to develop clean sediment totalmaximum daily loads �TMDLs� in incised streams and rivers�USEPA 1999�.

The Federal Interagency Stream Restoration Working Group�FISRWG� recently reviewed eight computer models frequentlyused in simulations of alluvial streams �FISRWG 1998�. All thesemodels exhibit in their present form one or more of the followinglimitations: �1� employ piecewise-steady inflow hydrographs andbackwater computations that are not suitable for flashy stream-flows in upland areas; �2� channel width adjustment through masswasting is either not considered or approximated through otherthan process-based algorithms; and �3� cannot simulate flow pro-cesses influenced by in-stream grade stabilization structures.Therefore, there is a definite need for a model that is free fromany of these restrictions.

In response to this need, the National Sedimentation Labora-tory of the U.S. Department of Agriculture-Agricultural ResearchService developed the CONCEPTS �CONservational ChannelEvolution and Pollutant Transport System� computer model toevaluate the long-term impact of DEC rehabilitation measures onsediment yield and TMDLs �Langendoen 2000�. CONCEPTS is aphysically based computer model capable of simulating processes
that shape degraded stream corridors over long periods of time.
URNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2008 / 749

This paper presents the model formulation of CONCEPTS andthe validation of flow and streambed evolution components. Twoaccompanying papers provide: �1� an extensive description andvalidation of the streambank erosion component; and �2� the ap-plication of the model to incised stream systems in Mississippi.

Evolution of Incised Streams

The evolution of incised stream systems has been widely studiedand the progression of the stream through various evolutionarystages has been documented in conceptual models of channel evo-lution �e.g., Schumm et al. 1984; Simon and Hupp 1986�.

In northern Mississippi, oversteepened reaches have developedin the channelized streams due to straightening and base-levellowering. Streams have responded through bed degradation, com-monly by upward migrating knickpoints or knickzones �Decour-sey 1981; Patrick et al. 1982; Schumm et al. 1984; Watson et al.1988�. Stream incision has been halted mainly by eroding intomore resistant lithologic units �Grissinger and Murphey 1983� orby grade control structures �Watson et al. 1997�, though bed deg-radation is still occurring in several stream systems even 40 yearsafter channelization �Simon 1998�.

The resulting oversteepening and overheightening of the stre-ambanks have led to a destabilizing of the banks and consequentcollapse predominantly by a slab-type failure mechanism �Littleet al. 1982�. After failure the bank is stable until failed materialis eroded from the toe of the bank, and the bank is again broughtto a limiting condition. Two- or threefold increases in channelwidth commonly have occurred after the passage of a knickpoint�Decoursey 1981; Patrick et al. 1982; Schumm et al. 1984�.

The increase in channel width has reduced the transport capac-ity of the streams, thereby promoting bed aggradation. Rates ofwidth increases eventually have slowed when the channels be-came so wide that the flow was unable to erode the failed bankmaterial. Invasive vegetation has established on the debris, soinducing deposition of fine sediment and encouraging the estab-lishment of a berm at the foot of the bank. Finally, the stream hasreturned to a quasiequilibrium state. Adjustment in this stagecommonly continues through increasing channel asymmetry, me-ander planform development, and meander growth �Thorne1999�. The adjustment process may span several decades, some-times exceeding 100 years.

Conceptual channel evolution models have aided in identify-ing and studying the processes playing a role in incised channeldynamics. However, these models are only able to qualitativelypredict several important phases in the evolution of incisedstreams, and they cannot predict the time it takes for streams toprogress through all those stages. Assessments and mitigation ofdownstream impacts demand accurate quantification of channel-erosion contribution to sediment yield. Enhanced numerical mod-els simulating all aspects of incised stream dynamics are thereforerequired.

Model Formulation

The main processes that must be simulated by a numerical modelof incised channel evolution are streamflow hydraulics, sedimenttransport, streambed evolution, and streambank erosion. This sec-tion discusses the conceptualization and mathematical character-
ization of these processes in CONCEPTS.
750 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2008

Hydraulics

Channel curvature of channelized, straightened streams is gener-ally small. Therefore, the flow may be modeled as one dimen-sional along the channel’s centerline. Berm and onset of meanderdevelopment in the late stage of quasiequilibrium recoveryrequire a multidimensional approach and cannot be simulatedby a physically and process-based one-dimensional model.CONCEPTS is currently being enhanced to address these pro-cesses. Nevertheless, the one-dimensional approach has a signifi-cant simulation advantage, in that it is computationally efficient tomodel the decadal temporal adjustment process.

Incision leads to reduced overland storage throughout the wa-tershed, increasing peak discharges of a given frequency and re-ducing time to peak �Shields and Cooper 1994�. Some streamscan hold their 100-year peak discharge. Hence, many incisedstream systems are ephemeral with flashy flow. This and the evo-lutionary nature of incised streams make a model based on achannel forming or effective discharge unsuitable.

In order to ensure an accurate unsteady-flow description,CONCEPTS uses a distributed flow routing scheme in which flowis computed as a function of time simultaneously at all sectionsalong the stream. Lumped flow routing schemes where the flow iscomputed as a function of time at one location along the stream,e.g. the Muskingum method �McCarthy 1938�, are computation-ally more efficient. However, they require the use of a uniformflow equation such as the Manning equation or a backwater tech-nique to derive flow depth. Use of piecewise-steady approxima-tions carries the basic implication that the upstream flood wavemigrates down the channel without experiencing any changes. Inreality, overbank storage, flow resistance, seepage, and large-scalelongitudinal mixing can all contribute to subsidence of the floodwave �i.e., wave diffusion and peak attenuation�. Further, lumpedflow routing schemes cannot accurately account for boundaryconditions imposed by grade control structures, culverts, bridgecrossings, or tributary inflows.

Basic Modeling EquationsUnsteady, gradually varying, one-dimensional open-channel flowis described by the Saint Venant equations �e.g., Chaudhry 1993�

B�y

�t+

�Q

�x= ql �1�

�Q

�t+

�

�x�Q2

A� + gA� �y

�x+ Sf� = 0 �2�

where Q=discharge; y=stage; A=flow area; B=flow top width;ql=lateral flow per unit length of channel into the channel �e.g.,overland flow or subsurface flow�, Sf =friction slope;g=gravitational acceleration; x=distance along the channel cen-terline; and t=time �see Fig. 1 for definition sketch�. The frictionslope is defined as

Sf =Q2

K2 =n2Q2

A2R4/3 �3�

where K=conveyance; R=hydraulic radius; and n=Manningroughness coefficient. In CONCEPTS, Manning n can vary overspace and time.

The approximation of the nonlinear convective accelerationterm �� /�x�Q2 /A�� may give rise to numerical difficulties. Theinertia terms of Eq. �2� can be neglected whenever: �1� convective
accelerations are small and discharge is increasing or decreasing

slowly in time; or �2� the Froude number �F� is small �i.e., F�0.1�. The resulting system of equations is known as the diffu-sion wave model �DiWM�, whereas Eqs. �1� and �2� are known asthe dynamic wave model �DyWM�. Both models are implementedin CONCEPTS. Whenever applicable, the diffusion wave modelis a more efficient and robust modeling choice and is preferred forsimulating long-term channel evolution with CONCEPTS. Thenext section shows that differences in results obtained by theDiWM and DyWM in CONCEPTS are generally small.

Numerical Solution MethodCONCEPTS uses the generalized Preissmann scheme to solve thegoverning system of equations. Some of the advantages of thePreissmann scheme are: �1� the scheme is simple, robust, compactand it has been extensively used; �2� the scheme allows a variablespatial grid; and �3� the scheme is implicit in time, which allowsfor large time steps. Many scientists �e.g., Cunge et al. 1980� havedocumented the numerical behavior of the method, involving sta-bility, convergence, and accuracy.

Any variable f and its derivatives in the generalized version ofthe original Preissmann scheme are approximated as:

f�x,t� � ��f j+1n+1 + �1 − ��f j

n+1� + �1 − ��f j+1n + �1 − ��f j

n�

�4a�

�f�x,t� � �

f j+1n+1 − f j+1

n

+ �1 − ��f j

n+1 − f jn

�4b�

Fig. 1. Definition sketch

�t �t �t

JO

�

�xf�x,t� � �

f j+1n+1 − f j

n+1

�x+ �1 − ��

f j+1n − f j

n

�x�4c�

where j=space index; n=time index; �t=time step; �x=spacestep; and � �0��1� and � �0��1�=temporal and spatialweighting factors, respectively, to provide model flexibility withrespect to stability and convergence. Setting �=1 /2 reduces Eqs.�4� to the original scheme of Preissmann �1961�.

Substituting Eq. �4� into the continuity and momentum equa-tions and applying upstream and downstream boundary condi-tions yield a nonlinear system of 2N equations in 2N unknowns,where N=number of computational sections. Rewriting the equa-tions in terms of differential changes of flow variables Q and ylinearizes the system of equations �for final form of discretizedequations see Langendoen 2000�. The resulting banded system ofequations is solved using the algorithm SGBSVX of theLAPACK software package for matrix computations �Anderson etal. 1999�. In practice this algorithm has shown to be more robustthan the widely used double sweep algorithm. The double sweepmethod may produce unrealistic changes in discharge and flowdepth at the beginning of runoff events �Langendoen 1997�.

The implementation of the solution method contains variousenhancements to improve the robustness of the model, particu-larly for flashy runoff events. Discharge increases rapidly fromnear dry-bed conditions after the beginning of rainfall to somepeak discharge, then falling to base flow again. The Preissmannscheme will predict unrealistic solutions of the dependent vari-ables in case of large increases of discharge and flow depth at the

rating model parameters
illust
upstream end of a computational section. Substituting the dis-


cretized momentum equation into the discretized continuity equa-tion to eliminate the differential change of discharge at thedownstream end of a computational section ��Qj+1�, yields thefollowing expression:

�yj+1 = a1�yj + a2�Qj �5�

where the differential change of a variable is defined as��·�= �·�n+1− �·�n, and a1 and a2=coefficients that depend onchannel geometry, flow, time and space steps, and spatial andtemporal weighting factors. When a fast-rising flood wave entersthe computational section at base flow, a1�−1, a2�0, anda2� �a1�. Therefore, the water surface elevation at the downstreamend of the computational section will dip and the predicted flowdepth hj+1 may become negative. This problem can be greatlyreduced by downwinding �0.5��1� the temporal derivative ofthe continuity equation. CONCEPTS tracks the front of a flood-wave to automatically adjust � thereby assuring positive flowdepths. Further, the time step is automatically adjusted to accu-rately represent temporal changes in discharge and flow depth. Inpractice the time step usually varies between 30 s and 10 min.

The approximation of the friction slope may also lead to nu-merical difficulties. The friction slope is nonlinear and a dominantterm in the momentum equation. French �1985� lists the four mosttypically used methods:

Arithmetic mean:

Sf =1

2�Sf j

+ Sf j+1� �6a�

Mean of conveyances and discharges:

Sf = �Qj + Qj+1

Kj + Kj+1�2

�6b�

Geometric mean:

Sf = Sf jSf j+1

�6c�

Harmonic mean:

Sf =Sf j

Sf j+1

Sf j+ Sf j+1

�6d�

Langendoen �1997� performed an extensive analysis of Eq.�6a�–�6d� based on an extension and enhancement of the analysisby Cunge et al. �1980�. The latter investigators analyzed the fric-tion slope models for a constant discharge, whereas Langendoen�1997� allowed a varying discharge, and introduced a spatialweighting of upstream and downstream discharges, conveyances,and friction slopes. Eq. �6a�–�6d� may predict friction slope val-ues that differ up to three orders of magnitude depending onwhether discharge and conveyance are increasing or decreasing instreamwise direction �Langendoen 1997�.

If conveyance is decreasing downstream �Kj+1�Kj�, Eqs. �6b�or �6c� are to be preferred. The arithmetic mean friction slope�Eq. �6a�� is of the order of the larger of Sf j

or Sf j+1, and is

therefore larger than the friction slopes predicted by Eqs. �6b� or�6c�. Consequently, the predicted upstream stage �yj� is largerwhen using Eq. �6a� than Eqs. �6b� or �6c�, see Fig. 2�a�. Notethat the difference in modeled water surface elevations displayedin Fig. 2�a� was reduced by the relatively small grid spacing im-mediately upstream of the knickpoint. If conveyance is increasingdownstream �Kj+1�Kj�, Eq. �6a� is to be preferred. For example,friction slope predicted by Eqs. �6b�–�6d� may produce rather
small or negative upstream flow depths �Fig. 2�b��. CONCEPTS

automatically switches between Eqs. �6a� and �6b� dependingwhether the ratio Kj /Kj+1 is smaller or larger than unity.

It was stated in the previous section that the hydraulics offlood waves simulated by the diffusion and dynamic wave modelsin CONCEPTS are fairly similar. To demonstrate, both modelswere used to simulate the propagation of various flood wavesthrough a 300 m wide and 25 km long rectangular channel. Theflow hydrograph at the channel inlet was represented by a prob-ability density function of the gamma distribution �Fig. 3�d��.Discharge at the channel inlet was varied between 1000 and2000 m3 /s; discharge peaked one hour after beginning of runoff.Three tests were carried out in which bed slope was varied togenerate diffusion and dynamic waves: �1� bed slope is0.008 m /m; �2� bed slope is 0.002 m /m; and �3� bed slope is0.0002 m /m. Using the criteria of Ponce et al. �1978�, the floodwaves can be classified as diffusion waves �Tests �1� and �2��and a dynamic wave �Test �3��. The numerical setup was:�x=500 m, �t=5 min, �=0.7, and �=0.5.

The simulated rating curves at the channel outlet are used to

Fig. 2. Effect of friction slope model on simulated water surfaceelevation: �a� conveyance decreasing in downstream direction above aknickpoint at 700 m; �b� conveyance increasing in downstreamdirection above a confluence; and �c� location map of modeledreaches on the Goodwin Creek Watershed, Miss.

compare the dynamics of the three flood waves simulated by the

diffusion and dynamic wave models �Figs. 3�a–c��. Fig. 3 showsthat both models simulate the increasing width of the ratingcurve’s loop with decreasing bed slope. As expected, the simu-lated rating curves for the diffusion waves are very similar �Figs.3�a and b��. Although, the rating curve of the dynamic wave simu-lated by the DiWM slightly differs from that simulated by theDyWM, calculated peak discharge and stage are similar �Fig.3�c��. In general, the peak discharge simulated by the DiWM atthe channel outlet is slightly smaller than that simulated by theDyWM: Test �1� 1,836 m3 /s versus 1,842 m3 /s; Test �2�1,616 m3 /s versus 1,624 m3 /s; and Test �3� 1,184 m3 /s versus1,192 m3 /s. In case of Test �3�, the width of the loop and there-fore the time difference between discharge and stage peaks islarger for the DyWM �Fig. 3�c��. This may affect the simulationof sediment transport mechanics, because the boundary shearstress calculated by the DyWM will be larger during the risinglimb and smaller during the falling limb of the hydrograph thanthat calculated by the DiWM �Fig. 3�d��.

Sediment Transport and Streambed Adjustment

Streambanks in the midsouth United States are typically com-posed of Holocene valley-fill deposits mainly containing clays,silts, and fine sands �Grissinger and Murphey 1983�. Streambedsare composed of sands and gravels or resistant clay layers, whichprovide most of the temporary bed control. Therefore, the range

Fig. 3. Comparison of diffusion and dynamic wave models used by Cchannel �300 m wide and 25 km long� are shown for varying bed slhydrograph specified at the inlet and calculated boundary shear stres

in particle sizes being transported in incised channels may be

JO

quite large and the composition of the sediment mixture in trans-port may be quite different from the bed-material compositionbecause a majority of the sediments are fines transported in sus-pension. The nonuniform sediment distribution should thereforenot be characterized by a representative sediment size. Sedimenttransport rates need to be calculated by size fraction.

CONCEPTS uses a total-load evaluation of bed-material trans-port and treats movement of clays and fine silts as pass-throughbackground washload. Although the mechanics of suspended andbed load movement are markedly different, combining them re-duces the number of equations to be solved and therefore thecomputational effort for long-term simulations. The differences intransport mechanics are accounted for through nonequilibriumeffects.

Basic Modeling EquationsThe conservation of sediment mass by size class reads

�Ack

�t+

�Sk

�x− pbk�b = qsk �7�

where ck=cross-sectional average sediment concentration of sizeclass k; Sk=total fractional load; S=�kSk=total bed-material load;pbk=fractional content of sediment particles of size class k on thestreambed ��kpbk=1�; and qsk=lateral input of sediment from up-

EPTS. Simulated rating curves at the inlet and outlet of a rectangular� 0.008 m /m; �b� 0.002 m /m,; and �c� 0.0002 m /m. The discharge

e outlet for a bed slope of 0.0002 m /m are shown in �d�.

ONCope: �as at th

land, tributary and streambank sources. The net flux of sediment


between streambed and water column �positive when directed up-ward� is computed as

�b = − �1 − ��Bbzb

�t�8�

in which =porosity of the bed; Bb=wetted width of the activestreambed; and zb=streambed elevation �Fig. 1�a��.

The entrainment rate of sediment particles from the bed differsfor cohesive and cohesionless bed material. For cohesionless stre-ambeds the entrainment rate is based on the formulation proposedby Bennett �1974� analogous to that of Foster and Meyer �1972�.The local erosion or deposition rate is proportional to the differ-ence between the sediment concentration �ck� and equilibriumsediment concentration �ck�

pbk�b =A

Tk�ck − ck� �9�

where Tk=time scale representing the adjustment rate of sedimentconcentration to equilibrium ��b=0�. Various formulations of Tk

exist in literature. For particles transported in suspension, Arma-nini and Di Silvio �1988� suggested

Tkk

h=

�s

h+ �1 −

�s

h�exp− 1.5��s

h�−1/6k

u*� �10�

where k=particle fall velocity; �s=thickness of the surfacelayer; and u

*=shear velocity. Formulations for sediment particles

transported as bed load have also been determined, e.g., Phillipsand Sutherland �1989�. However, Eq. �10� is employed for all sizeclasses in CONCEPTS. Though not derived for bed load, Eq. �10�correctly tends to ck= ck for coarse particles in which caseTk��x /u, where u=flow velocity.

The transport, deposition, and erosion processes of cohesivesediments is extremely complex due to their highly varying prop-erties and, therefore, behavior when their environment changes.Erosion rate is most commonly given by a linear excess-shearstress formulation in the form developed at the University of Cali-fornia at Davis �e.g., Ariathurai and Arulanandan 1978�

�b = BbM� b

c− 1� �11�

where M =erosion-rate coefficient; b=bed shear stress, and c=critical shear stress above which bed material is entrained.The properties of the cohesive bed material are lumped into theparameters M and c. These parameters can be measured in situusing portable flumes or submersible jet devices. Hanson �1990�developed a jet testing device that was used by Hanson andSimon �2001� to measure M and c in the midcontinentalUnited States. The deposition rate is given by Krone’s �1962�formulation

− pbk�b = Bbkck�1 − b

c,d,k� �12�

where c,d,k=shear stress below which particles in transport beginto deposit.

For graded bed material the sediment transport rate depends onthe bed-material composition which itself depends on historicalerosion and deposition rates. Following Hirano �1971� the stre-ambed is divided into a surface layer and one or more substratelayers �Fig. 1�a��.

The sediment mass balance of the surface layer is


�1 − ��Aspsk

�t= − pbk�b + �sk �13�

where As=area of surface layer; psk=fractional content of sedi-ment particles of size class k in the surface layer ��kpsk=1�; and�sk=net flux of sediment between surface layer and substrate�positive when directed upward�

�sk = − �1 − ��sk

�Bb�zb − �s��t

�14�

where

�sk = ��pbk + �1 − ��psk if �sk � 0

p0k if �sk � 0 �15�

in which p0k=fractional content of sediment particles of size classk in the substrate ��kp0k=1�, and � �0��1� is an exchangeparameter. The previous mixing model of �sk in case of aggrada-tion ��sk�0� was proposed by Hoey and Ferguson �1994�. Vary-ing � enables the simulation of downstream fining ��1� and acoarser surface layer overlaying a finer substrate ��0�.

The thickness of the surface layer is associated with the timescale under consideration �Rahuel et al. 1989�. The surface layerthickness has been considered to be a function of dune height orwater depth �Armanini and Di Silvio 1988; Rahuel et al. 1989�,or grain size �Borah et al. 1982; van Niekerk et al. 1992; Cui et al.1996�. In CONCEPTS the thickness of the surface layer is set tothe larger of 10% of the flow depth or the D90 of the surface layer,where D90�90th percentile of the grain size distribution. Rahuelet al. �1989� propose a thickness of the surface layer between 10and 20% of the flow depth, whereas Armanini and Di Silvio�1988� suggest a minimum thickness of 5% of the flow depth.

Layers within the substrate have a maximum, user-specifiedthickness. If this thickness is exceeded a new layer is added.CONCEPTS keeps track of the composition of all substrate lay-ers. A sediment balance, similar to Eq. �13�, is solved for the toplayer.

The equilibrium sediment concentration or sediment transportcapacity �ck� is computed using a modification of the sedimenttransport capacity predictor SEDTRA, originally developed byGarbrecht et al. �1995�. SEDTRA calculates the total sedimenttransport capacity by size fraction for twelve predefined sizeclasses �0.01�D�50 mm, where D=particle diameter� with asuitable transport equation for each size fraction: Laursen �1958�for silts, Yang �1973� for sands, and Meyer-Peter and Mueller�1948� for gravels. SEDTRA accounts for the interdependence be-tween size fractions for initiation of motion by a critical sedimentdiameter for each size fraction:

Dc,k = Dk�Dm

Dk��

�16�

where Dm=geometric mean size of the bed material and�=hiding coefficient �0��1�. For �=1, the mean size of thesediment is the critical diameter for all size fractions and all frac-tions tend to move at the same flow strength. For �=0, each sizefraction behaves independently of the others and the Dk of eachsize fraction is used to calculate initiation of motion.

Garbrecht et al. �1995� tested SEDTRA against Brownlie’s�1981� data set. About 80% of the computed laboratory data arewithin a factor of 2 of the measured values, and about 90% withina factor of 3. About 55% of the computed field data are within a
factor of three, and about 70% within a factor of 5.

Numerical Solution MethodEq. �7� can be written as

�Ack

�t+

�uAck

�x− pbk�b = qsk �17�

and is solved for Ack, which is the transported sediment masswithin a cross section. Eq. �17� is solved using a fractional

erosion of cohesive streambanks as a combination of: �1� lateral

JO

step method �Yanenko 1971�, yielding the equivalent system ofequations

�Ack

�t+

�uAck

�x= qsk �18a�

dAck

dt= �

A

Tk�ck − ck� for cohesionless streambed material

BbM� b

c− 1� − kck�1 −

b

c,d,k�� for cohesive streambed material � �18b�

A fractional step method is necessary because of the disparatetime scales of advection and sediment entrainment/deposition forcoarse sediment particles, that is �x /uTk�1. For coarse sedimentparticles, the source term is dominant, which may lead to unstableor inaccurate solutions �e.g., LeVeque and Yee 1990�.

The advection equation �18a� is solved using the method ofcharacteristics by backtracing the characteristic in the x– t planefrom point �xj+1 , tn+1� to its intersect with t= tn �C�1� or x=xj

�C�1�:

�Ack� j* = � �1 − C��Ack� j

n + C�Ack� j−1n + qsk�t for C � 1

�1 −1

C��Ack� j−1

n+1 +1

C�Ack� j−1

n + qsk�t for C � 1 ��19�

where C=u�t /�x=Courant number, and the intermediate solu-tion �Ack�* is used as initial condition for the initial value problem�Eq. �18b��. If streambed material is cohesionless, Eq. �18b� issolved analytically by assuming constant Tk and ck over a timestep �t:

�Ack�n+1 = Ack + ��Ack�* − Ack�exp�− �t/Tk� �20�

If streambed material is cohesive

�Ack�n+1 = �Ack�* + �tpbk�b �21�

The change in bed elevation is then obtained from Eq. �8� as

�zb = �k

�zbk = − �k

1 + C

�1 − �Bb��Ack�n+1 − �Ack�*� �22�

The composition of the surface layer is computed from Eq. �13�as

�Aspsk�n+1 = �Aspsk�n + Bb�zbk − �skBb��zb − ��s� �23�

Bank Erosion and Stream Width Adjustment

In-depth descriptions of the formulation and field validation ofthis model component are reported by Langendoen and Simon�2008�. A brief outline of the treatment of bank erosion is pre-sented herein for the sake of completeness. CONCEPTS treats

erosion of the bank toe by fluvial entrainment of bank-materialparticles, often termed hydraulic erosion; and �2� mass failure ofthe bank due to gravity.

Hydraulic ErosionThe rate of lateral erosion of cohesive bank-material is equivalentto that of cohesive bed-material �Eq. �11��

E = M� sb

c− 1� �24�

where E=lateral erosion rate and sb=shear stress exerted onbank material. The erodibility of bank soils can be increasedmarkedly by weakening processes such as weathering. The pro-cesses responsible for loosening and detaching grains and aggre-gates are closely associated with soil moisture conditions at andbeneath the bank surface, which may be impacted by vegetation.Swelling and shrinking of soils during repeated cycles of wettingand drying can contribute to cracking that significantly increaseserodibility and reduces soil shear strength. On the other hand,vegetation can also increase the erosion resistance of the soil.

The lateral erosion distance over a simulation time step �t isE�t �Fig. 1�b��. An average erosion distance is computed for eachlayer comprising the composite bank material. The average shearstress exerted by the flow on each soil layer is computed usingeither the vertical depth or the normal area method �Lundgren andJonsson 1964�. In spite of some shortcomings associated withthese models, they are adopted in CONCEPTS because of theirsimplicity. Bank material is removed over the erosion distanceand added as lateral sediment inflow into the channel �qsk�, as-suming immediate disaggregation of the material into its primaryparticles.

Values of c can be obtained from: �1� Arulanandan et al.�1980� if sodium adsorption ratio, dielectric dispersion, and porefluid salt concentration are known; �2� in situ measurements �Tol-hurst et al. 1999; Hanson and Simon 2001�; or �3� historical dataon the retreat of the base of the bank combined with flow data.The effects of weathering processes and vegetation can be in-cluded by adjusting c. The predicted rate of lateral erosion issensitive to the values used for critical shear stress and erosion-rate coefficient or erodibility of the bank material. Critical shearstress and erodibility may vary greatly both spatially and tempo-rally, e.g., due to variations in soil water. Values obtained from in
situ measurements may therefore be unreliable, and should be

collected systematically along the bank profile and over a longerperiod, hence providing a range of values that can be used toperform an uncertainty analysis.

Streambank StabilityStreambank failure occurs when gravitational forces that tend tomove soil downslope exceed the forces of friction and cohesionthat resist movement. The risk of failure is usually expressed by afactor of safety �FS� representing the ratio of resisting to drivingforces or moments. Banks may fail by four distinct types of fail-ure mechanisms �ASCE 1998�: �1� planar failures; �2� rotationalfailures; �3� cantilever failures; and �4� piping and sappingfailures.

Steep banks commonly fail along planar failure surfaces, withthe failure block sliding downward and outward into the channelbefore toppling. High, mildly sloped streambanks �bank angleless than 60°� usually fail along curved surfaces; the block offailed material tends to rotate towards the bank as it slides. Can-tilevered or overhanging banks are generated when erosion of anerodible layer in a stratified bank leads to undermining of over-lying, erosion-resistant layers. Streambanks may also fail byexfiltrating seepage and internal erosion known as piping andsapping �Hagerty 1991�. CONCEPTS performs stability analysesof planar and cantilever failures, which are most widely observedin the incised stream systems of the midcontinental United States.The bank’s geometry, soil properties, pore-water pressures, con-fining pressure exerted by the water in the stream, and riparianvegetation determine the stability of the bank.

Planar failures are analyzed using the limit equilibriummethod developed for engineered slopes and embankments �e.g.,Fredlund and Krahn 1977�. This method computes the requiredshear strength �or mobilized shear strength� to maintain a condi-tion of limiting equilibrium. The potential failure block is dividedinto slices �Fig. 1�c��, and the forces and moments acting on themare analyzed to determine if the block remains stationary. Divid-ing up the soil mass into a number of slices allows one to accom-modate differing slide mass geometries, stratified soils within themass and external loads such as trees. The stability analysis inCONCEPTS satisfies vertical force equilibrium for each slice andthe overall horizontal force equilibrium.

The resulting FS for the entire block reads �Langendoen2000�:

FS =sec �� j

�Ljcj� + Nj tan � j� − Uj tan � jb�

tan �� jWj − Fw

�25�

where �=failure plane angle; j=slice index; Lj =length of theslice base; cj�=effective cohesion; Nj =normal force on the base ofthe slice; � j�=effective angle of internal friction; Uj =pore-waterforce on the base of the slice; � j

b=angle indicating the increase inshear strength for an increase in matric suction �negative Uj�;Wj =weight of the slice; and Fw=horizontal component of thehydrostatic force exerted by the surface water on the streambank�Figs. 1�a and b��. The bank stability analysis assumes hydrostaticpore-water pressure distribution in both the saturated and unsat-urated portion of the failure block. The approximated pore-waterpressure may differ significantly from the actual pore-water pres-sure in case of large hydraulic gradients and near the soil surfacewhere pore-water pressure is greatly affected by infiltration andevapotranspiration processes. Research is ongoing to simulate
streambank hydrology �Langendoen et al. 2005�.

It should be noted that Eq. �25� yields FS as a function of theunknown failure plane angle �. A modified quadratic fitting pro-cess based on the search method of Kaufman et al. �1995� is usedto search for the minimum FS corresponding to �min. This search-ing process converges very rapidly to �min, minimizing the num-ber of evaluations of factor of safety for a given failure blockgeometry.

Banks often have a composite structure incorporating materi-als of noncohesive and cohesive nature. Noncohesive sandygravel deposits formed from relic bars are commonly overlain bycohesive sandy silt and clays deposited by overbank flow onemergent bars. Larger shear stresses exerted by the flow on thebase of the bank and higher erodibility of the noncohesive bankmaterial leads to undercutting of the streambank and the forma-tion of cantilevers.

Three principal modes of cantilever failure have been recog-nized: Shear, beam, and tensile failure �Thorne and Tovey 1981�.CONCEPTS analysis of cantilever stability is limited to shearfailure, and the corresponding factor of safety for shear failure�FSs� is determined as the ratio of the shear strength of the bankmaterials and the weight of the cantilever block, resulting in

FSs =� j

�Ljcj� − Uj tan � jb�

� jWj

�26�

where j=index of bank-material layers in the failure block.When the factor of safety computed for a given streambank

block becomes less than one the bank is assumed to undergo massfailure and the volume of failed streambank material is added aslateral sediment inflow into the channel �see Eq. �7��, assumingimmediate disaggregation of the material into its primaryparticles.

Boundary Conditions

External boundary conditions are imposed at the channel inlet andoutlet, whereas internal boundary conditions are specified at tribu-tary inflows and in-stream structures. Sieben �1997� described indetail the number and type of conditions required at externalboundaries of mobile-bed models in order to have a well-posedproblem �e.g., p. 152 of Hirsch 1988�. In case of subcritical flow,following the recommendations of Sieben �1997�, discharge hy-drograph and sediment transport rate are imposed at the upstreamboundary, and water level or a flow rating curve is specified at theoutlet. Sediment transport rates can be imposed as: �1� time se-ries, Sk=Sk�t�; �2� a rating curve, Sk=Sk�Q�; or �3� as a fraction ofsediment transport capacity, Sk=Qck. The flow rating curve,Q=Q�h�, at the outlet node can be user supplied, or CONCEPTScan calculate a loop-rating curve based on local flow conditionsQ=Q�h ,�y /�x�=K�h�Sf��y /�x�.

Characterization of In-Stream Hydraulic StructuresConventional flow routing models do not usually include in-stream hydraulic structures. However, structures can act as con-trols or transitions having their own local rating curves and canchange the state of flow at a point compared to the same flowwithout structures. Four types of structures are incorporated in thecurrent version of CONCEPTS: culvert, bridge crossing, dropstructure, and a generic structure. The latter can be any structurefor which the rating curve is available, e.g., a measuring flume.

To guarantee an efficient solution method, the mathematical
representation of the flow rate at instream hydraulic structures has

to be equivalent to that of the channel flow, that is Eqs. �1� and�2�. CONCEPTS employs a continuity and dynamic equation forhydraulic structures, which establish a relation between dis-charges and stages upstream and downstream of the structure. Thecontinuity equation states that discharge across the structure isconstant, that is

�Q

�x= 0 �27�

The dynamic equations for each hydraulic structure type are givenby Langendoen �2000�.

To seamlessly incorporate the approximations to the governingequations of hydraulic structures into the solution method, theseapproximations must be written in terms of differential changes ofthe dependent variables

�Q+ − �Q− = Q−n − Q+

n �28�

�Q+ −�fQ

�y+�y+ −

�fQ

�y−�y− = fn − Q+

n �29�

where fQ represents the dynamic equation of the structure, and thesubscripts � and � denote cross sections upstream and down-stream of the structure, respectively.

The derivatives of fQ with respect to y+ and y− can be analyti-cally derived. However, if the structure acts as a control, that iswhen the flow at the structure is critical, this approach is not veryrobust. In this case the derivative of fQ with respect to y+ isapproximated as

�fQ

�y+=

2�Q

y+�Q + �Q� − y+�Q − �Q��30�

where the function y+�Q�=headwater equation and �Q=presetdischarge increment.

It is assumed that sediment will not deposit in or on structures.The part of the sediment mass carried in suspension will be trans-ported through or over the structure. The part of the sedimentmass transported as bed load will be adjusted for the elevation ofthe upstream invert of the structure. The bed load will be depos-ited upstream of the structure if the upstream invert is locatedabove the streambed, otherwise it will pass the structure. Thefractional bed load is calculated from the total fractional loadbased on the ratio u

*/k �Julien 1995�.

Validation

The adjustment of incised stream systems comprises two mainmorphological changes: streambed evolution �degradation andaggradation�, and channel widening. In this section, the capabilityof CONCEPTS to simulate streambed adjustments are testedagainst laboratory and field experiments reported in the technicalliterature.

Degradation

Ashida and Michiue �1971� carried out laboratory experimentsto study the degradation of a graded stream bed and the develop-ment of an armor layer in case of clear water inflow. Theymeasured sediment transport rates by grain size, variations ingrain-size distribution of the armor layer, and changes in bed
elevation with time. The dataset reported for Run 6 of these ex-
JO

periments was used to examine the capability of CONCEPTS toaccurately quantify degradation rates and composition of aneroded streambed.

The experimental flume was 0.8 m wide and 20 m long. Waterdischarge was 0.0314 m3 /s, Manning n was 0.02, and initial bedslope was 0.01. The sediment supply rate at the upstream end waszero, and the elevation of the bed at the downstream end wascontrolled with a weir. Particle diameter of the bed materialranged from 0.1 to 10 mm. The grain-size distribution had a geo-metric mean of 1.4 mm and a geometric standard deviation of−1.7� �where D in millimeters=2−��. The numerical setup was:�x=1 m, =0.4, and �=0.7 for all size fractions.

Fig. 4�a� plots the simulated depth of degradation with thatmeasured at sections 7, 10, and 13 m upstream of the weir. Tocompare the simulated and measured values, an exponential-rise-to-max function of the form

E = a1�1 − exp�− a2t�� 31�

was fitted through the simulated and measured data. In Eq. �31�, Eis depth of erosion and a1,2 are coefficients. The coefficient a1

represents the final depth of erosion �E��, the product a1�a2

represents the initial rate of erosion. The r2 value of the regres-sions varied between 0.99 and 1.0.

Fig. 4�a� shows that the rate of erosion is underpredicted. Thedifference between simulated and observed initial erosion ratevaries between 75% for the 7 m section �0.9 versus 3.4 cm /h� to93% for the 13 m section �1.0 versus 13.5 cm /h�. However, thesimulated final depth of erosion agrees well with that observed.Generally, simulated E� along the flume is about 0.3 cm largerthan that observed. For the 7 m section, simulated E� is 7% larger�4.6 versus 4.3 cm� than that observed. For the 10 m section,simulated E� is 5% larger �6.1 versus 5.8 cm�. For the 13 m sec-tion simulated E� is 6% larger �7.3 versus 6.9 cm�.

Figs. 4�b and c� show the variation of surface layer composi-tion with time at sections 10 and 13 m upstream of the weir,respectively. CONCEPTS predicts both the rate of developmentand the final grain-size distribution of the armor layer quite well.Though, the simulated sand-sized fractions, particularly between0.5 and 2 mm, are slightly overpredicted. However, sand-sizedsediment particles and very fine gravel are still being entrainedfrom the armor layer after 100 h of simulation. At the 10 m sec-tion simulated geometric mean diameter �3.8 mm� and geometricstandard deviation �−1.1�� deviate 15% and 0.2� from thosemeasured �4.5 mm and −0.9�, respectively�. At the 13 m section,simulated geometric mean diameter �4.3 mm� is 4% smaller thanthat measured �4.4 mm�, whereas simulated geometric standarddeviation �−1.0�� is 0.2� smaller than that measured �−0.8��.

The degradation rate is underpredicted mainly because thetransport capacity computed by SEDTRA in the first few hours ofthe experiment is too small. The measured sand-sized transportrates are about two times larger, whereas the measured gravel-sized transport rates are about four times larger. The entrainmentof sediment particles smaller than 4 mm in the later stages ofarmor layer development is probably underpredicted because ofthe selected value of the hiding coefficient ��=0.7�. This valuewas selected following criteria proposed by Wilcock �1993�, andsensitivity runs did not result on significant degradation changeswith �. However, with the reduction in bed slope �reduced shearstress and consequently Shields number� and development of thearmor layer �geometric mean diameter increased from sand-sizedto gravel-sized�, the value of the hiding coefficient may be toolarge, thereby reducing sediment transport rates for particles
smaller than the geometric mean diameter. This notwithstanding,

final erosion depths and grain-size distribution of the streambedare quite closely approximated by CONCEPTS. Wu �2004� showsthat changing bed roughness when the bed is being scoured canyield greater degradation rates.

Aggradation

To study downstream fining of sediment researchers at the SaintAnthony Falls Laboratory �SAFL�, University of Minnesota, car-ried out a series of experiments that produced self-formed, longi-tudinally sorted gravel deposits �Paola et al. 1992; Seal et al.1997; Toro-Escobar et al. 2000�. During each experiment watersurface and bed profiles were measured. At the end of each ex-periment the bed surface and subsurface sediments were sampled.The resulting comprehensive data set provides for a detailedtesting of CONCEPTS capabilities to simulate bed profileand surface and subsurface composition of aggrading streams.CONCEPTS was tested against Run 2 of the SAFL experiments.Details of the experimental setup and measured data are found inSeal et al. �1997�.

Fig. 4. Comparison of simulation results and measurements of Runof erosion at 7, 10, and 13 m upstream of the downstream end; �b� vaand �c� variation of surface-layer composition 13 m upstream of the

The experimental channel was 60 m long, 1.2 m deep, and


0.3 m wide. Water discharge, downstream flow depth, and sedi-ment feed rate were kept constant during the run at 0.049 m3 /s,0.45 m, and 5.65 kg /min, respectively. Sediment was fed manu-ally 1 m downstream of the headgate. The feed material waspoorly sorted and weakly bimodal; geometric mean size was4.6 mm and geometric standard deviation was −2.6�. The nu-merical setup was: �x=2 m, n=0.025, =0.35, �=0.9 for thesand-sized fractions and �=0.7 for the gravel-sized fractions.Best results were obtained for ��0.3, which is smaller than thevalue ��=0.7� backcalculated by Toro-Escobar et al. �1996� andused by Cui et al. �1996� in their model simulations. These au-thors excluded the suspended load and therefore did not accountfor the sand-sized fractions, although the deposit contained about30% sand �Seal et al. 1997�.

Fig. 5 compares results of the simulations with measurementsof: �1� bed and water surface profiles; �2� grain size statistics ofbed surface and subsurface; and �3� distribution of the sand frac-tion within the deposit. The simulated evolution of the bed profileand final water surface profile agree well with those observed

e degradation experiments by Ashida and Michiue �1971�: �a� depthof surface-layer composition 10 m upstream of the downstream end;tream end

6 of thriationdowns

�Fig. 5�a��. The simulated average velocity of the gravel front is

1.25 m /h, which is 7% larger than that observed �1.17 m /h�. Thesimulated average height of the gravel front is 0.21 m, which is0.01 m smaller than that observed.

The simulated geometric mean of the grain-size distribution ofthe deposit and surface layer agrees very well with that measured�Fig. 5�b��. Both longitudinal �downstream� and vertical fining ofthe deposit is accurately simulated. The simulated geometric stan-dard deviation is slightly underpredicted by about 0.3�. Thedownstream and vertical changes in percent sand are accuratelysimulated within the deposit, but percent sand is approximately

Fig. 5. Comparison of simulation results and measurements of Run 2�b� grain size statistics of bed surface and subsurface; and �c� disrepresented by solid lines, whereas measured data are plotted as symsampling the substrate between lines representing the location of the

6% too large in the surface layer �Fig. 5�c��.

JO

These results show that CONCEPTS can accurately simulatethe evolution of bed profile and surface and subsurface grain-sizestatistics for an aggrading channel.

Summary

The channel model CONCEPTS was developed to: �1� simulatethe long-term evolution of incised channel systems; and �2� evalu-ate the effectiveness of stream restoration designs. It simulates thethree main processes that shape incised streams: hydraulics, sedi-

downstream fining experiments by Seal et al. �1997�: �a� bed profiles;on of the sand fraction within the deposit. Simulation results areubsurface grain-size statistics and percent sand were determined by

rofile at the times indicated in the legend.

of thetributibols. Sbed p

ment transport and bed adjustment, and streambank erosion.


The flow routing method used in CONCEPTS allows largetime steps and enhancements incorporated into the solutionmethod make CONCEPTS very robust and efficient for long-termsimulations. The major enhancements are: �1� automatically ad-justing the spatial weighting coefficient � to stabilize the propa-gation of flood waves over nearly dry beds, when large variationsin dependent variables lead to a breakdown of the generalizedPreissmann scheme; and �2� automatic switching between frictionslope models depending on water surface profiles, e.g. those cre-ated by converging channel flow �e.g., controls� and divergingchannel flow �e.g., backwater�.

The sediment transport component was designed to simulatethe wide range of sediment particle sizes transported through in-cised streams. CONCEPTS calculates total load bed-materialtransport for 14 predefined size classes ranging from clay andvery fine silt to small cobbles over both cohesive and cohesion-less streambeds. The differences in transport mechanics of thevarious size classes are accounted for through nonequilibriumeffects.

The streambank erosion submodel simulates the two processesresponsible for the retreat of cohesive streambanks: hydraulic ero-sion and gravitational mass failure. Bank material may comprisevarious soil units. The bank stability analysis accounts for thedestabilizing effects of positive pore-water pressures, the increasein shear strength due to matric suction, and the confining pres-sures exerted by the water in the stream.

CONCEPTS has been validated against laboratory and fieldexperiments. A comparison against measured data from Ashidaand Michiue �1971� and Seal et al. �1997� show that streambedelevation and grain-size distribution of surface and subsurfacestreambed material of degrading and aggrading streams are accu-rately simulated. Langendoen et al. �1999� and Langendoen andSimon �2008� show that the long-term retreat of the right bank ofa reach on the Goodwin Creek, Mississippi, was accurately simu-lated by CONCEPTS. Results from these validation studies dem-onstrate CONCEPTS ability to simulate the dynamics of incisedstreams. An accompanying paper �in preparation� presents the ap-plication of CONCEPTS to two channelized stream systems innorthern Mississippi.

CONCEPTS was designed to use data that can be either ob-tained in situ or from literature. The data required is similar toother widely used one-dimensional sediment transport models:channel geometry, boundary roughness, water and sediment in-flows, and resistance to erosion �i.e., grain size distribution orcritical shear stress�. The bank stability analysis requires the fol-lowing data: soil shear strength �cohesion and friction angle�,bulk soil weight, and elevation of phreatic surface. The model,workshop materials, and sample data can be obtained from thefirst writer.

Acknowledgments

The development of CONCEPTS was supported by theUSDA-Agricultural Research Service. Francisco Simões andWeiming Wu kindly reviewed this work and made many helpfulsuggestions.

Notation

The following symbols are used in this paper:A � flow area �m2�;

As � area of surface layer �m2�;−1
a1,2 � coefficients �m and s , respectively�;

B � top width �m�;C � Courant number;c � sediment concentration;c � equilibrium sediment concentration �transport

capacity�;c� � effective cohesion �N /m2�;D � particle diameter �mm�;E � lateral erosion rate �m/s� or depth of erosion �m�;F � Froude number;

Fw � hydrostatic force �N/m�;FS � factor of safety;

f � variable;fQ � dynamic equation of a hydraulic structure;g � gravitational acceleration �m /s2�;h � flow depth �m�;K � conveyance �m3 /s�;L � length �m�;

M � erosion-rate coefficient �m/s�;N � normal force �N/m�;n � Manning roughness coefficient �s /m1/3�;p � sediment size fraction;Q � discharge �m3 /s�;ql � lateral flow per unit channel length �m2 /s�;qs � lateral sediment input �m2 /s�;R � hydraulic radius �m�;S � sediment load �m3 /s�;

Sf � friction slope;T � adjustment rate of sediment concentration �s�;t � time �s�;

U � pore-water force �N/m�;u � flow velocity �m/s�;

u* � shear velocity �m/s�;x � streamwise coordinate �m�;y � stage �m�;z � elevation �m�;� � size fraction of sediment exchanged between

surface layer and substratum;� � failure plane angle;

�t ,�x � time �s� and space �m� intervals;� � thickness �m�;� � hiding coefficient;

� ,� � temporal and spatial weighting factors; � porosity; � shear stress �N /m2�;

� � sediment flux �m2 /s�;� � log2 D, where D is in millimeters;

�� angle of internal friction;�b � angle indicating the increase in shear strength for

an increase in matric suction;� � exchange parameter; and � particle fall velocity �m/s�.

Superscripts and Subscripts

b � streambed;c � critical;j � space or slice index;k � size class;n � time index;s � surface layer;

sb � streambank;0 � substrate; and

�,� � upstream and downstream sections of a hydraulic
structure.

References

Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra,J., Du Croz, J., Greenbaum, A., Hammerling, S., McKenney, A., andSorensen, D. �1999�. LAPACK users’ guide, 3rd Ed., Society for In-dustrial and Applied Mathematics �SIAM�, Philadelphia.

Ariathurai, R., and Arulanandan, K. �1978�. “Erosion rates of cohesivesoils.” J. Hydr. Div., 104�2�, 279–283.

Armanini, A., and Di Silvio, G. �1988�. “A one-dimensional model forthe transport of a sediment mixture in nonequilibrium conditions.” J.Hydraul. Res., 26�3�, 275–292.

Arulanandan, K., Gillogley, E., and Tully, R. �1980�. “Development of aquantitative method to predict critical shear stress and rate of erosionof natural undisturbed cohesive soils.” Rep. No. GL-80-5, U.S. ArmyCorps of Engineers, Waterways Experiment Station, Vicksburg, Miss.

ASCE Task Committee on Hydraulics, Bank Mechanics, and Modeling ofRiver Width Adjustment. �1998�. “River width adjustment. I: Pro-cesses and mechanisms.” J. Hydraul. Eng., 124�9�, 881–902.

Ashida, K., and Michiue, M. �1971�. “An investigation of river bed deg-radation downstream of a dam.” Proc., 14th Congress of the Int. As-sociation for Hydraulic Research, C30-1-9.

Bennett, J. P. �1974�. “Concepts of mathematical modeling of sedimentyield.” Water Resour. Res., 10�3�, 485–492.

Borah, D. K., Alonso, C. V., and Prasad, S. N. �1982�. “Routing gradedsediments in streams: Formulations.” J. Hydr. Div., 108�12�, 1486–1503.

Brownlie, W. R. �1981�. “Compilation of alluvial channel data: Labora-tory and field.” Rep. No. KH-R-43B, W. M. Keck Laboratory of Hy-draulics and Water Resources, California Institute of Technology,Pasadena, Calif.

Carling, P. A. �1988�. “The concept of dominant discharge applied to twogravel-bed streams in relation to channel stability threshold.” EarthSurf. Processes Landforms, 13�4�, 355–367.

Chaudhry, M. H. �1993�. Open-channel flow, Prentice-Hall, EnglewoodCliffs, N.J.

Cooper, C. M., Hudson, F. D., and Bray, K. �1996�. “The demonstrationerosion control �DEC� project in the Yazoo Basin.” Proc., 6th FederalInteragency Sedimentation Conf., Las Vegas, 43–46.

Cui, Y., Parker, G., and Paola, C. �1996�. “Numerical simulation of aggra-dation and downstream fining.” J. Hydraul. Res., 34�2�, 185–204.

Cunge, J. A., Holly, F. M., Jr., and Verwey, A. �1980�. Practical aspectsof computational river hydraulics, Pitman, Boston.

Decoursey, D. G. �1981�. “Channel stability.” Rep. to the Corps of Engi-neers, Vicksburg District under Section 32 Program, Work Unit, 7,U.S. Dept. of Agriculture, Agricultural Research Service, NationalSedimentation Laboratory, Oxford, Miss.

Dunne, T., and Leopold, L. B. �1978�. Water in environmental planning,Freeman, San Francisco.

Federal Interagency Stream Restoration Working Group �FISRWG�.�1998�. Stream corridor restoration: Principles, processes, and prac-tices, U.S. Dept. of Agriculture, Natural Resources Conservation Ser-vice, Washington, D.C.

Foster, G. R., and Meyer, L. D. �1972�. “A closed-form soil erosionequation for upland areas.” Sedimentation Symp. to Honor ProfessorHans Albert Einstein, H. W. Shen, ed., Colorado State University, FortCollins, Colo., 12-1–12-19.

Fredlund, D. G., and Krahn, J. �1977�. “Comparison of slope stabilitymethods of analysis.” Can. Geotech. J., 14�3�, 429–439.

French, R. H. �1985�. Open-channel hydraulics, McGraw-Hill, NewYork.

Garbrecht, J., Kuhnle, R. A., and Alonso, C. V. �1995�. “A sedimenttransport capacity formulation for application to large channel net-works.” J. Soil Water Conservat., 50�5�, 527–529.

Grissinger, E. H., and Murphey, J. B. �1983�. “Present channel stabilityand late Quaternary valley deposits in northern Mississippi.” Modernand ancient fluvial systems, J. D. Collinson and J. Lewin, eds., Black-
well Scientific Publications, Oxford, U.K., 241–250.
JO

Hagerty, D. J. �1991�. “Piping/sapping erosion. I: Basic considerations.”J. Hydraul. Eng., 117�8�, 991–1008.

Hanson, G. J. �1990�. “Surface erodibility of earthen channels at highstresses. Part II—Developing an in-situ testing device.” Trans. ASAE,33�1�, 132–137.

Hanson, G. J., and Simon, A. �2001�. “Erodibility of cohesive streambedsin the loess area of the midwestern USA.” Hydrolog. Process., 15�1�,23–38.

Hirano, M. �1971�. “River bed degradation with armoring.” Proc., JapanSociety of Civil Engineers, 195, 55–65.

Hirsch, C. �1988�. Numerical computation of internal and external flows,Wiley, Chichester, U.K.

Hoey, T. B., and Ferguson, R. �1994�. “Numerical simulation of down-stream fining by selective transport in gravel bed rivers: Model devel-opment and illustration.” Water Resour. Res., 30�7�, 2251–2260.

Julien, P. Y. �1995�. Erosion and sedimentation, Cambridge UniversityPress, Cambridge, U.K.

Kaufman, E. H., Jr., Leeming, D. J., and Taylor, G. D. �1995�. “AnODE-based approach to nonlinearly constrained minimax problems.”Numer. Algorithms, 9�1�, 25–37.

Krone, R. B. �1962�. “Flume studies of the transport of sediment in es-tuarine shoaling processes.” Final Rep., Hydraulic Engineering Labo-ratory, Univ. of California, Berkeley, Calif.

Langendoen, E. J. �1997�. “DWAVNET: Modifications to friction slopeformulation and solution method.” Technical Rep. No. CCHE-TR-97-1, Center for Computational Hydroscience and Engineering, Univ.of Mississippi, University, Miss.

Langendoen, E. J. �2000�. “CONCEPTS—Conservational channel evolu-tion and pollutant transport system.” Research Rep. No. 16, U.S. Dept.of Agriculture, Agricultural Research Service, National SedimentationLaboratory, Oxford, Miss.

Langendoen, E. J., Lowrance, R. R., Williams, R. G., Pollen, N., andSimon, A. �2005�. “Modeling the impact of riparian buffer systems onbank stability of an incised stream.” Proc., World Water and Environ-mental Resources Congress, Anchorage, Alaska.

Langendoen, E. J., and Simon, A. �2008�. “Modeling the evolution ofincised streams. II: Streambank erosion.” J. Hydraul. Eng., 134�7�,2008.

Langendoen, E. J., Simon, A., Curini, A., and Alonso, C. V. �1999�.“Field validation of an improved process-based model for streambankstability analysis.” 1999 Int. Water Resources Engineering Conf.�CD-ROM�, R. Walton and R. E. Nece, eds., ASCE, Reston, Va.

Laursen, E. �1958�. “The total sediment load of streams.” J. Hydr. Div.,84�1�, 1–36.

LeVeque, R. J., and Yee, H. C. �1990�. “A study of numerical methods forhyperbolic conservation laws with stiff source terms.” J. Comput.Phys., 86�1�, 187–210.

Little, W. C., Thorne, C. R., and Murphey, J. B. �1982�. “Mass bankfailure analysis of selected Yazoo Basin streams.” Trans. ASAE,25�5�, 1321–1328.

Lundgren, H., and Jonsson, I. G. �1964�. “Shear and velocity distributionin shallow channels.” J. Hydr. Div., 90�1�, 1–21.

McCarthy, G. T. �1938�. “The unit hydrograph and flood routing.” Conf.North Atlantic Div., U.S. Corps of Engineers, New London, Conn..

Meyer-Peter, E., and Mueller, R. �1948�. “Formula for bed-load trans-port.” Proc., Int. Association for Hydraulic Research, 2nd Meeting,Stockholm, Sweden.

Paola, C., Parker, G., Seal, R., Sinha, S. K., Southard, J. B., and Wilcock,P. R. �1992�. “Downstream fining by selective deposition in a labora-tory flume.” Science, 258�5089�, 1757–1760.

Patrick, D. M., Smith, L. M., and Whitten, C. B. �1982�. “Methods forstudying accelerated fluvial change.” Gravel-bed rivers, R. D. Hey, J.C. Bathurst, and C. R. Thorne, eds., Wiley, Chichester, U.K., 783–816.

Phillips, B. C., and Sutherland, A. J. �1989�. “Spatial lag effects in bedload sediment transport.” J. Hydraul. Res., 27�1�, 115–133.

Ponce, V. M., Simons, D. B., and Li, R.-M. �1978�. “Applicability of


kinematic and diffusion models.” J. Hydraul. Res., 104�3�, 353–360.Preissmann, A. �1961�. “Propagation des intumescences dans les canaux

et rivières.” Proc., 1st Congress of the French Association for Com-putation, Grenoble, France, 433–442 �in French�.

Rahuel, J. L., Holly, F. M., Chollet, J. P., Belleudy, P. J., and Yang, G.�1989�. “Modeling of riverbed evolution for bedload sediment mix-tures.” J. Hydraul. Eng., 115�11�, 1521–1542.

Schumm, S. A., Harvey, M. D., and Watson, C. C. �1984�. Incised chan-nels: Morphology, dynamics and control, Water Resources Publica-tions, Littleton, Colo.

Seal, R., Paola, C., Parker, G., Southard, J. B., and Wilcock, P. R. �1997�.“Experiments on downstream fining of gravel. I: Narrow-channelruns.” J. Hydraul. Eng., 123�10�, 874–884.

Shields, F. D., Jr., and Cooper, C. M. �1994�. “Riparian wetlands andflood stages.” Proc., Hydraulic Engineering ’94, G. V. Controneo andR. R. Rumer, eds., ASCE, New York, 351–355.

Sieben, J. �1997�. “Modelling of hydraulics and morphology in mountainrivers.” Communications on Hydraulic and Geotechnical EngineeringRep. No. 97-3, Faculty of Civil Engineering, Delft Univ. of Technol-ogy, Delft, The Netherlands.

Simon, A. �1998�. “Processes and forms of the Yalobusha River System:A detailed geomorphic evaluation.” Research Rep. No. 9, U.S. Dept.of Agriculture, Agricultural Research Service, National SedimentationLaboratory, Oxford, Miss.

Simon, A., Dickerson, W., and Heins, A. �2004�. “Suspended-sedimenttransport rates at the 1.5-year recurrence interval for ecoregions of theUnited States: Transport conditions at the bankfull and effective dis-charge?” Geomorphology, 58�1–4�, 243–262.

Simon, A., and Hupp, C. R. �1986�. “Channel evolution in modified Ten-nessee channels.” Proc., 4th Federal Interagency SedimentationConf., Las Vegas, 5-71–5-82.

Thorne, C. R. �1999�. “Bank processes and channel evolution in the in-cised rivers of North-Central Mississippi.” Incised river channels:Processes, forms, engineering and management, S. E. Darby and A.Simon, eds., Wiley, Chichester, U.K., 97–121.

Thorne, C. R., and Tovey, N. K. �1981�. “Stability of composite river


banks.” Earth Surf. Processes Landforms, 6�5�, 469–484.Tolhurst, T. J., Black, K. S., Shayler, S. A., Mather, S., Black, I., Baker,

K., and Paterson, D. M. �1999�. “Measuring the in situ erosion shearstrength of intertidal sediments with the Cohesive Strength Meter�CSM�.” Estuarine Coastal Shelf Sci., 49�2�, 281–294.

Toro-Escobar, C. M., Paola, C., Parker, G., Wilcock, P. R., and Southard,J. B. �2000�. “Experiments on downstream fining of gravel. II: Wideand sandy runs.” J. Hydraul. Eng., 126�3�, 198–208.

Toro-Escobar, C. M., Parker, G., and Paola, C. �1996�. “Transfer functionfor the deposition of poorly sorted gravel in response to streambedaggradation.” J. Hydraul. Res., 34�1�, 35–53.

U.S. Environmental Protection Agency �USEPA�. �1999�. “Protocol fordeveloping sediment TMDLs.” EPA 841-B-99-004, Office of Water,Washington, D.C.

van Niekerk, A., Vogel, K., Slingerland, R. L., and Bridge, J. S. �1992�.“Routing of heterogeneous sediments over movable bed: Model de-velopment.” J. Hydraul. Eng., 118�2�, 246–262.

Watson, C. C., Harvey, M. D., Biedenharn, D. S., and Combs, P. G.�1988�. “Geotechnical and hydraulic stability numbers for channelrehabilitation. Part 1: The approach.” Proc., Hydraulic Engineering,J. Gessler and S. R. Abt, eds., ASCE, New York, 120–125.

Watson, C. C., Raphelt, N. K., and Biedenharn, D. S. �1997�. “Historicalbackground of erosion problems in the Yazoo Basin.” Management oflandscapes disturbed by channel incision, S. S. Y. Wang, E. J.Langendoen, and F. D. Shields, Jr., eds., The University of Missis-sippi, University, Miss, 115–119.

Wilcock, P. R. �1993�. “Critical shear stress of natural sediments.” J.Hydraul. Eng., 119�4�, 491–505.

Wu, W. �2004�. “Depth-averaged two-dimensional numerical modeling ofunsteady flow and nonuniform sediment transport in open channels.”J. Hydraul. Eng., 130�10�, 1013–1024.

Yanenko, N. N. �1971�. The method of fractional steps, Springer, NewYork.

Yang, C. T. �1973�. “Incipient motion and sediment transport.” J. Hydr.Div., 99�10�, 1679–1704.

Documents

Modeling the Evolution of Incised Streams: I. Model ... · Modeling the Evolution of Incised Streams: I. Model Formulation and Validation of Flow and Streambed Evolution Components