Page 1

The Unified Gravel-Sand (TUGS) Model: Simulating

Sediment Transport and Gravel/Sand Grain Size

Distributions in Gravel-Bedded Rivers

Yantao Cui1

Received 9 July 2006; revised 7 August 2007; accepted 14 August 2007; published 30 October 2007.

[1]

transport, erosion, and deposition of both gravel and sand. TUGS model employs the

surface-based bed load equation of Wilcock and Crowe (2003) and links grain size

distributions in the bed load, surface layer, and subsurface with the gravel transfer function

of Hoey and Ferguson (1994) and Toro-Escobar et al. (1996), a hypothetical sand transfer

function, and hypothetical functions for sand entrainment/infiltration from/into the

subsurface. The model is capable of exploring the dynamics of grain size distributions,

including the fractions of sand in sediment deposits and on the channel bed surface, and is

potentially useful in exploring gravel-sand transitions and reservoir sedimentation

processes. Simulation of three sets of large-scale flume experiments indicates that the

model, with minor adjustment to the Wilcock-Crowe equation, excellently reproduced bed

profile and grain size distributions of the sediment deposits, including the fractions of

sand within the deposits. Simulation of a flushing flow experiment indicated that the sand

entrainment function is potentially capable of simulating the short-term processes such

as flushing flow events.

This paper presents The Unified Gravel-Sand (TUGS) model that simulates the

Citation: Cui, Y. (2007), The Unified Gravel-Sand (TUGS) Model: Simulating Sediment Transport and Gravel/Sand Grain Size

Distributions in Gravel-Bedded Rivers, Water Resour. Res., 43, W10436, doi:10.1029/2006WR005330.

1.Introduction

[2] Understanding the dynamics of grain size distribu-

tions, particularly the fractions of fine sediment (sand and

finer) in channel bed deposits in salmonid bearing rivers is

of grave importance. Adult salmonids select locations with

favorable hydraulic conditions and appropriate grain size

distributions to deposit their eggs, which generally incubate

for a period of about two to five months [Beacham and

Murray, 1990]. In addition to egg mortality due to expo-

sures from redd scour during flood events, two other

potential risks for incubating salmonid eggs are low survival

rate due to low intragravel flow and entombment of fry,

both of which are usually the result of high fine sediment

content in the spawning habitats [e.g., Coble, 1961; Cooper,

1965; Phillips et al., 1975]. To date, only a few numerical

sediment transport models attempted to predict the evolu-

tion of sand fraction in a gravel deposits. For example,

Ferguson [2003] explored the emergence of abrupt gravel-

sand transitions in rivers while Wu and Chou [2003]

explored the effect of flushing flow with a numerical model;

both models include gravel and sand. The models of

Ferguson [2003] and Wu and Chou [2003] divided sediment

into gravel and sand fractions while no detailed grain size

distributions of either gravel or sand was simulated. In

particular, the model of Wu and Chou [2003] applied a

sediment transport equation that resembles the two-fraction

sediment transport equation of Wilcock [1998], while the

model of Ferguson applied a sediment transport equation

with similar concept as the two-fraction equation of Wilcock

and Kenworthy [2002]. Few numerical models capable of

simulating both gravel and sand are currently available

because the interaction between sediment deposits and

sediment particles in transport (bed load) is an extremely

complex process, which is poorly understood, especially

when both fine and coarse sediments are considered. It can

be expected that the fraction of sand in a sediment deposit is

positively correlated with sand supply, as implemented in

Wu and Chou [2003]. However, other factors may signifi-

cantly affect the deposition of sand in a sediment deposit of

gravel-sand mixture, whether it is framework-supported or

matrix-supported. Cui and Parker [1998], for example,

suggested that the fractions of sand in sediment deposits

of gravel-sand mixtures are highly correlated to the standard

deviation of the gravel class of the sediment deposit: a

sediment deposit composed only of coarse sediment with a

smaller standard deviation means more uniform sediment

particles, which implies more pore space available for fine

sediment to infiltrate. It can be expected that a numerical

model that describes the dynamics of gravel grain size

distributions is potentially capable of addressing the con-

cerns of Cui and Parker’s [1998]. In addition, a model

describing the grain size distributions of gravel will also

allow for inclusion of particle abrasion, which can be

critically important in modeling long-river reaches because

particle abrasion accelerates the transport of sediment, as

demonstrated by Cui and Parker [2005]. Most of the current

fractional-based sediment transport models for gravel bed-

1Stillwater Sciences, Berkeley, California, USA.

Copyright 2007 by the American Geophysical Union.

0043-1397/07/2006WR005330$09.00

W10436

WATER RESOURCES RESEARCH, VOL. 43, W10436, doi:10.1029/2006WR005330, 2007

Click

Here

for

Full

Article

1 of 16

Page 2

ded rivers treat fine sediment (sand and finer) as throughput

load, thus excluding it from the simulation. An example of

such a model is the gravel pulse model of Cui and Parker

[2005], which applies the surface-based bed load equation

of Parker [1990], an equation that excludes sand and finer

particles. To include sand in the simulation of sediment

transport, erosion, and deposition processes following re-

moval of dams, Cui and Wilcox [2007] and Cui et al.

[2006a, 2006b] assumed that gravel and sand transport by

different processes (bed load versus suspended load) and at

different timescales (years versus days). They further as-

sumed that gravel and sand transport are weakly correlated

and can be assumed to be independent of each other, thus

allowing for application of their respective equations inde-

pendently. The treatment of Cui and Wilcox [2007] and Cui

et al. [2006a, 2006b] allowed for a simple evaluation of

potential sand deposition over a gravel bed in the absence of

a unified gravel-sand transport equation. The models of Cui

and Wilcox [2007] and Cui et al. [2006a, 2006b], however,

cannot be used for predicting subsurface sand fractions in

the absence of a relation linking sand fractions in bed load,

the surface layer and the subsurface.

[3] The recent Wilcock and Crowe [2003] sediment

transport equation provides the first sediment transport

relation that calculates both gravel and sand transport on a

fractional-basis that accounts for the effect of surface sand

fraction on particle mobility. A formulation is proposed

herein as a first-order approximation, linking sand fractions

in interface sediment (i.e., sediment to become part of

subsurface during aggradation, and sediment to be eroded

from subsurface during degradation) and the surface layer

(more details about the surface, subsurface and interface are

provided below in Section 2). In addition, hypothetical

relations are proposed to calculate the entrainment of sand

from the subsurface and infiltration of sand into the sub-

surface based on the concept for upward sand movement

proposed by Wilcock et al. [1996] and Wu and Chou [2003].

Combined with Wilcock and Crowe’s [2003] sediment

transport equation and a gravel transfer function proposed

by Hoey and Ferguson [1994] and Toro-Escobar et al.

[1996], the proposed formulations were incorporated into

The Unified Gravel-Sand (TUGS) model. The model is then

applied to simulate three relatively large-scale flume experi-

ments conducted at St. Anthony Falls Laboratory (SAFL)

and reported by Paola et al. [1992], Seal et al. [1995, 1997],

and Toro-Escobar et al. [1996]. Without any modification to

the coefficients in the equation of Wilcock and Crowe’s

[2003], the model excellently reproduced the grain size

distributions of the sediment deposits. The simulated bed

slopes for all the three runs, however, are steeper than that

observed in the experiments. Several attempts are made to

improve the simulated bed slope, and it was found that

replacing the dimensionless sediment transport ? normal-

ized shear stress relation in the Wilcock and Crowe [2003]

equation (i.e., Equation (7) in the original reference) with a

Parker [1990] type of relation matches both bed slopes and

grain size distributions of the sediment deposit for all the

three runs. This adjustment is considered to be minor, and

the excellent match between simulation and observation

with a minor adjustment indicate that the model is likely to

be useful in simulating natural and management scenarios in

rivers. In order to explore the potential usefulness of the

entrainment function, the flushing flow experiment of Wu

and Chou [2003] is simulated and produced reasonable

results compared with the observed data. In a manuscript

submitted concomitantly with this manuscript [Cui, 2007], I

examine model performance under field scale and provide

comparisons of bed material fine sediment fractions under

different hydrologic and sediment supply conditions.

2.

for Grain Size Distributions

[4] The conceptual model adapted in TUGS model is the

widelyusedthree-layermodel[e.g.,Hirano,1971;Ribberink,

1987; Parker, 1990, 1991a, 1991b; Parker and Sutherland,

1990; Wilcock and Crowe, 2003; Cui et al., 2003b, 2006a,

2006b; Cui and Parker, 2005; Cui and Wilcox, 2007].

According to this conceptual model, a sediment deposit in

a gravel bedded river is composed of a surface layer (or

active layer), which lies on top of the subsurface sediment

(the second layer). A third bed load-layer is composed of

the sediment particles transported as bed load over the

surface layer. The three layers, along with the concept of

the interface layer, are shown in Figure 1. For the simulation

of channel bed dynamics, the interface layer was defined by

previous researchers [e.g., Hirano, 1971; Ribberink, 1987;

Parker 1991a, 1991b) as the layer of sediment to be

deposited on top of the existing subsurface layer during

aggradation (Figure 1a) and the layer of sediment to be

released from the top of the subsurface layer during degra-

dation (Figure 1b). That is, interface sediment becomes part

of the subsurface layer following aggradation and was part

of the subsurface layer prior to channel degradation. Note

that the interface layer exists only conceptually because,

given a time increment Dt, the thickness of the interface

layer during this time increment approaches zero when Dt

! 0, and hence the conceptual model is conventionally

named as a three-layer model instead of a four-layer model.

The basic concept of the bed load, surface, subsurface, and

interface layers will help the understanding of the surface-

based bed load equation and the sediment exchange func-

tions presented in detail below. The grain size distribution of

the surface layer sediment will be part of the input variables

for applying the Wilcock and Crowe’s [2003] bed load

equation. Here it is important to first introduce the notations

that describe the grain size distributions in the four layers

before the Wilcock and Crowe’s [2003] equation and other

relations are introduced.

[5] To describe the grain size distribution of a bulk of

sediment, it is first divided into two classes: sediment

coarser than 2 mm (gravel and coarser, which will be

referred to as gravel hereafter), and sediment finer than

2 mm (sand and finer, which will be referred to as sand

hereafter). For simplicity, it is assumed that there is no

sorting within the sand class during its transport except that

its fraction may vary at different locations and change in

time. Thus the grain size distribution for the sand class can

be simplified as a geometric mean grain size Dgsand a

geometric standard deviation sgs, where the first subscript g

denotes geometric and the second subscript s denotes sand.

Procedures for calculating Dgsand sgscan be found in the

Conceptual Three-Layer Model and Notations

2 of 16

W10436

CUI: TUGS MODEL

W10436

Page 3

work of Cui et al. [1996]. The gravel class is divided into N

groups bounded by N + 1 grain sizes, D1, D2, ..., DN, DN+1,

where D1is always 2 mm (i.e., the boundary between sand

and gravel). The j-th size group, where j is between 1 and N,

is bounded by grain size Djand Dj+1with a mean grain size

Dj¼

class (i.e., the bulk sediment excludes sand and finer) is then

represented with volumetric fractions of the N size groups.

Knowing the fraction of sand within the bulk sample, and

the fraction of each gravel size group within the gravel

class, we will be able to define the grain size distribution of

the combined gravel and sand. For example, we would

describe the grain size distribution of the surface layer with

sand fraction Fs, and the fractions of different gravel size

groups within the gravel class, F1, F2, ..., FN, so that F1, F2,

..., FNsum to unity. It is important to note that the notation

for grain size distributions used here is different from that

by Wilcock and Crowe [2003] due to the simplification

made to treat the entire sand class as one grain size group,

and yet, to preserve its geometric mean and geometric

standard deviation values. The above notation allows for

more concise presentations of gravel and sand transfer

functions (to be presented later) than if the notations of

Wilcock and Crowe [2003] is adapted. The simplification of

treating the entire sand class as a single bin is necessary

because it significantly reduces the computer memory

during simulation, thus allowing for storage of more sedi-

ment deposit layers. Because sand class is simplified as a

single bin, the same geometric mean grain size (Dgs) and

geometric standard deviation (sgs) apply to sand class in all

the sediment used in the model (i.e., in bed load, surface

layer, subsurface, and interface). Similar to the surface layer,

where the fraction of sand and the fractions of different

gravel size groups within the gravel class are denoted as [Fs

and (F1, F2, ..., FN)], the fraction of sand and fractions of

different gravel size groups within the gravel class for bed

load, subsurface, and interface are denoted as [ps, (p1, p2,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

DjDjþ1

p

. The grain size distribution of the gravel

..., pN)], [fs, (f1, f2, ..., fN)], and [fIs, (fI1, fI2, ..., fIN)],

N

pj= 1,

respectively, where

X

j¼1

X

N

j¼1

fj= 1, and

X

N

j¼1

fIj=1.

3.

Combined Coarse and Fine Sediment [Wilcock

and Crowe, 2003]

[6] TUGS model implements the surface-based bed load

equation of Wilcock and Crowe [2003] for evaluation of bed

load transport capacity. Here only a brief summary of the

equation is provided to facilitate the discussions of the

model, and its details can be found in the original reference

[Wilcock and Crowe, 2003]. As pointed out earlier, some of

the notations used in this paper are different from the

original notations by Wilcock and Crowe [2003], and thus

certain components of the Wilcock and Crowe [2003]

equation given below may have a different form than the

original reference. The Wilcock and Crowe [2003] equation

adapted with the notations used in this paper is presented

below:

Surface-Based Bed Load Equation for

Wj*¼

0:002 t=trj

?

14 1 ? 0:894=

j ¼ 1;2;...;N; or s

in which t denotes bed shear stress, trsand trj(j = 1, 2,...,

N) denote the reference shear stress for sand and for the j-th

size group of the gravel class, respectively (to be discussed

in more detail below), and Ws* and Wj* denote dimension-

less transport rate for sand and for the j-th size group of the

gravel class, as defined below:

?7:5;

for t=trj< 1:35

ffiffiffiffiffiffiffiffiffiffi

t=trj

p

??4:5;

for t=trj? 1:35

(

;

ð1Þ

W*

s¼RgQs

Bu3

*Fs

;

ð2aÞ

W*

j¼

RgQgj

*1 ? Fs

Bu3

ðÞFj

ð2bÞ

Figure 1. Conceptual three-layer model in a gravel bedded river [after Parker 1991a, 1991b], showing

the bed load layer, the surface layer, and the subsurface layer. The interface sediment is the sediment that

becomes part of the subsurface during aggradation (a) and the subsurface sediment to be eroded from the

subsurface during degradation (b).

W10436

CUI: TUGS MODEL

3 of 16

W10436

Page 4

in which R denotes submerged specific weight of sediment

particles, g denotes gravitational acceleration, B denotes

channel width (assuming a rectangular channel), u*denotes

shear velocity (and u*=

ffiffiffiffiffiffiffiffi

and Qgjdenotes the volumetric transport rate for the j-th size

group of the gravel class. The reference shear stresses, trs

and trj, were defined by Wilcock and Crowe [2003] so the

shear stress at which the dimensionless sediment transport

rates, Ws*and Wj*, have a low value of 0.002, as indicated

in equation (1). The reference shear stress, trsand trj, are

grain size dependent:

t=r

p

, where r denotes the density

of water), Qsdenotes the volumetric transport rate for sand,

trs

trsg

¼

Dgs

Dsg

??b

;

ð3aÞ

trj

trsg¼

Dj

Dsg

??b

;

ð3bÞ

b ¼

0:67

1 þ exp 1:5 ? Dj=Dsg

??

ð3cÞ

in which trsg denotes a surface geometric mean based

reference shear stress, and Dsgdenotes the geometric mean

grain size of the surface layer (including both sand and

gravel classes). On the basis of flume experimental data,

Wilcock and Crowe [2003] proposed that surface geometric

mean based reference shear stress, trsg, decreases with the

increase in surface sand fraction (Fs):

trsg

rRgDsg

¼ 0:21 þ 0:15exp ?20Fs

ðÞð4Þ

The above equations allow for the calculation of the

transport rates of sand (Qs) and gravel at each size group

(Qgj), and from which, to calculate the overall gravel

transport rate (Qg) and the bed load grain size distribution as

expressed with values of ps, p1, p2, ..., pN:

Qg¼

X

N

j¼1

Qgj;

ð5aÞ

ps¼

Qs

Qsþ Qg

;

ð5bÞ

and pj¼Qgj

Qg

ð5cÞ

4.

Subsurface Sediment Deposit With the Surface

Layer

[7] The Wilcock and Crowe’s [2003] equation does not

provide a direct link between grain size distributions in the

bed load and the surface layer with that in the subsurface.

Linking Grain Size Distributions of the

Before the Wilcock and Crowe’s [2003] equation can be

implemented into a numerical model, a linkage between the

bed load, surface layer and subsurface grain size distribu-

tions needs to be established.

[8] The transfer of sediment among the bed load, surface

layer and subsurface are discussed for cases of bed degra-

dation and bed aggradation below. In the case of bed

degradation, it has been recognized since the work of

Hirano [1971] that the surface layer mines the subsurface

(Figure 1b), and thus,

fIj¼ fj;

ð6aÞ

and fIs¼ fs

ð6bÞ

Equation (6a) has been used for simulation of bed

degradation in all the Parker family of models [e.g., Parker,

1991a, 1991b; Cui et al., 1996, 2003b, 2006a, 2006b; Cui

and Parker, 2005; Cui and Wilcox, 2007]. Note that

Equation (6b) implies that fine sediment in the deposit

cannot be entrained unless the bed is eroded. Flume

observations, however, indicate that, although fine sediment

in the deposit cannot be entrained while the surface layer is

static, it can be entrained to a depth of up to 2 to 3 surface

layer thickness once the surface layer is mobilized [e.g.,

Diplas and Parker, 1985]. The entrainment of fine sediment

without bed scouring is discussed further following the

discussion of gravel and sand transfer functions in cases of

bed aggradation below.

[9] In case of bed aggradation, Cui and Parker [2005]

and Cui et al. [2003b, 2006a, 2006b] all applied the

formulation proposed by Hoey and Ferguson [1994]:

fIj¼ cpjþ 1 ? c

ðÞFj

ð7Þ

in which c = 0.7 was derived based on the St. Anthony

Falls Laboratory (SAFL) downstream fining experiment

Run 3 by Toro-Escobar et al. [1996]. It should be noted that

Toro-Escobar et al. [1996] excluded sediment particles finer

than 2 mm from their analysis because the sand fraction data

cannot collapse to the same relation as the gravel class

sediments. With that, Equation (7), which, together with

Equation (6a), will be referred to as gravel transfer function

hereafter, will be implemented to TUGS model for the

gravel class sediment. Here it is important to reiterate that

interface sediment is the sediment that works into the

subsurface layer during bed aggradation (Figure 1a) or to

be released from the subsurface during bed degradation

(Figure 1b). That is, any given layer of subsurface sediment

is the integration of the interface sediment over a period of

time during which this particular layer of subsurface

sediment was deposited, and thus, subsurface sediment

samples can be used as a surrogate of interface sediment in

deriving sediment transfer functions [e.g., Toro-Escobar et

al., 1996]. Similar to the practice of Toro-Escobar et al.

[1996], subsurface sediment samples will be used as

surrogates for interface sediment when a hypothetical sand

4 of 16

W10436

CUI: TUGS MODEL

W10436

Page 5

transfer function for the case of bed aggradation is

compared with field data below.

[10] To simplify the sand transfer relation, bed load sand

fraction is dropped out of the relation, and only the fractions

of sand in the surface layer and interface are considered.

Because the surface and bed load sand fractions are strongly

correlated, as suggested by the Wilcock and Crowe [2003]

equation, linking the subsurface sand fraction only to the

surface sand fraction implicitly links the subsurface sand

fraction to the bed load sand fraction. The following

considerations and constraints are taken into account in

formulating a hypothetical sand transfer function during

channel aggradation: (1) The interface sediment sand frac-

tion should increase as the surface sand fraction increases;

(2) the interface sediment sand fraction should decrease

with the increase in the geometric standard deviation of the

gravel class of the interface sediment, because a higher

geometric standard deviation implies less matrix space left

for sand deposition [Cui and Parker, 1998]; (3) the sand

fraction in the interface sediment should be equal to or

higher than surface sand fraction because the subsurface

sediment should generally be finer than that in the surface

layer; and (4) the calculated range of the interface sand

fractions should generally fall within the same range as the

subsurface sand fractions measured in the field. The pro-

posed hypothetical sand transfer function during bed aggra-

dation is written as

fIs¼

0:4 ? 0:1sggþ 0:6 þ 0:1sgg

Fs;

??Fs;

for sgg< 4

for sgg? 4:

?

ð8Þ

in which sggdenotes geometric standard deviation of the

gravel class of the interface sediment. Equation (8) is

constructed so that it satisfies the four considerations and

constraints discussed earlier. In addition, fIsapproaches to

unit as Fsincreases to unity. Note that fls6¼ 0 when Fs= 0,

which seems to be a violation of the physical principle that

no sand should be deposited into the subsurface if there is

no sand in transport (indicated with Fs= 0). While it is

seemingly the case, the physical principle is guaranteed in

the mass conservation calculation in the simulation, in

which the rate of fine sediment deposition cannot exceed the

rate of sediment transport at the upstream node, thus

satisfying the condition that fIs = 0 when Fs = 0. The

threshold of sgg= 4 in Equation (8) is chosen so that it

satisfies fIs ? Fs. As a matter of fact, very rarely the

geometric standard deviation of the gravel portion of a

sediment deposit exceeds 4. The predicted interface sand

fraction with the hypothetical sand transfer function

(Equation (8)) for different gravel geometric standard

deviations are shown in Figure 2, in comparison with field

measurements of subsurface sand fractions. Figure 2a shows

that the interface sand fraction increases with the increase in

surface sand fraction and decreases with the increase in

gravel geometric standard deviation. In addition, the field

measurements of subsurface sand fractions fall in the

general range of the predicted interface sand fractions. In

Figure 2b, interface sand fractions are calculated based on

the observed surface sand fractions and subsurface gravel

geometric standard deviations, and compared with mea-

Figure 2.

aggradation, compared with field and flume data for

surface/subsurface sand fractions. (a) hypothetical relation

under different subsurface geometric mean standard devia-

tions, in comparison with field data; (b) calculated interface

sand fraction presented as a function of surface sand

fraction, in comparison with field measurement of subsur-

face sand fractions; (c) calculated interface sand fraction

presented as a function of subsurface gravel standard

deviation, in comparison with field measurements of

subsurface sand fractions. Data source: Clear Creek: Matt

Brown, Jess Newton, Graham Matthews and Associates

[2003a, 2004]; other rivers in North California (Sacramento,

Trinity, Stanislaus, Tuolumne, and tributaries to Trinity):

Graham Matthews and Associates [2001, 2003b], Geoff

Hales, CDWR [1994, 1995]; McKenzie watershed: John

Wooster (unpublished data).

Hypothetical sand transfer function during

W10436

CUI: TUGS MODEL

5 of 16

W10436

Page 6

sured subsurface sand fractions. The comparison shows that

the calculated interface sand fractions and measured

subsurface sand fractions generally fall into the same range.

Figure 2c shows the same data as Figure 2b, except that

they are presented as a function of gravel geometric

standard deviation, showing the decreasing interface/subsur-

face sand fraction with increasing gravel geometric standard

deviation.

5.

From/Into the Subsurface

[11] The gravel and sand transfer functions discussed

above simulate the co-erosion and co-deposition of gravel

and sand in case of bed degradation and aggradation. Fine

sediment within the deposit, however, can be entrained for

up to a depth of two to three times of the surface layer

thickness without bed scouring and redeposition once the

surface layer is mobilized [Diplas and Parker, 1985] in the

absence of bed aggradation and degradation. In addition, a

sediment deposit that does not have enough fine sediment in

its pores (e.g., shortly after a flushing flow event) will allow

fine sediment to infiltrate back into the subsurface, if fine

sediment is available in bed load or suspended load. Many

flume experiments have suggested that the infiltration depth

is usually a few surface layer thicknesses [e.g., Beschta and

Jackson, 1979; Diplas and Parker, 1985], which is similar

to the depth of fine sediment entrainment. In the absence of

physically based equations to describe the entrainment and

infiltration of fine sediment from and into the subsurface

deposits, hypothetical relations based on the concept of

Wilcock et al. [1996] and Wu and Chou [2003] are proposed

below and implemented in the model. Before introducing

the hypothetical relations, the concepts for equilibrium

surface sand fraction for sand entrainment (Fse) and equi-

librium subsurface sand fraction for sand infiltration (fse) are

introduced. The equilibrium surface sand fraction for sand

entrainment is a surface sand fraction value, above which no

subsurface sand will be entrained. If Equation (8) is con-

sidered as an equilibrium relation between surface sand

fraction and subsurface sand fraction, it can be reversed to

come up with the equilibrium surface sand fraction for sand

entrainment by replacing interface sand fraction fIs with

subsurface sand fraction fs:

Entrainment and Infiltration of Fine Sediment

Fse¼

fsþ 0:1sgg? 0:4

fs;

??= 0:6 þ 0:1sgg

??;

for sgg< 4

for sgg? 4

?

ð9Þ

The equilibrium subsurface sand fraction for sand infiltration

is a subsurface sand fraction, above which no sand will be

able to infiltrate into the subsurface deposit. Note that

subsurface sand fraction can be higher than this equilibrium

subsurface sand fraction through co-deposition of gravel and

sand, realized through the implementation of Equation (8) in

the numerical model. The equilibrium subsurface sand

fraction for sand infiltration is defined based on visual fitting

of field and laboratory data shown in Figure 2c, and is given

below:

fse¼ 1:8exp ?0:85sgg

??

ð10Þ

The sand entrainment and infiltration fluxes per unit area are

then hypothesized as:

qe¼

aeu3

*

RgDsg

0;

W*

it=trsg

??Fse? Fs

ðÞ;

for Fs< Fse

for Fs? Fse:

8

:

<

ð11aÞ

qi¼

aiv3

RgDsg

0;

s

Fsfse? fs

ðÞ;

forfs< fse

for fs? fse

8

:

<

ð11bÞ

in which qeand qidenote sand entrainment and infiltration

fluxes per unit area, respectively; vs denotes settling

velocity, and aeand aiare dimensionless coefficients that

must be assigned through model calibration. Note that both

qeand qihave units of velocity. The term (aeu*

Equation (11a) is proposed in analogy to the sediment

transport equations of Parker [1990] and Wilcock and

Crowe [2003], and in Equation (11b) shear velocity u*is

replaced with settling velocity vs. The implication of

Equation (11a) is that the rate of entrainment is proportional

to sand transport rate and proportional to the deviation of

surface sand fraction from its equilibrium value. The

implication of Equation (11b) is that the rate of fine

sediment infiltration is proportional to surface fine sediment

fraction and the deviation of subsurface sand fraction from

its equilibrium value. An overall vertical sand transport rate

per unit channel area, qvs, can be calculated by combining

the fluxes of entrainment and infiltration:

3)/(RgDsg) in

qsv¼ qe? qi

ð12Þ

Note that qsvcan be either positive or negative. A positive

qsvvalue indicates a net upward sand flux (i.e., entrain-

ment) and a negative qsvvalue indicates a net downward

sand flux (i.e., infiltration).

[12] It is important to reiterate that the concepts used in

Equations (11a) and (11b) in this section are hypothetical,

and can only be used with adequate model calibration. The

equations, however, can be conveniently replaced with

better ones once they are available. In addition, fine

sediment infiltration and entrainment into/from subsurface

can be assumed to cancel off each other for long-term

simulations of natural rivers (i.e., by setting ai= 0 and

ae= 0). For the moment, they are implemented in TUGS

model for exploratory purposes. Later in this paper the

flushing flow data of Wu and Chou [2003] are used to

demonstrate that the entrainment function, calibrated to a

specific problem, can produce reasonable results.

6.Governing Equation for Water Flow

[13] The governing equations for water flow used in this

model are identical to that in Cui et al. [2006a, 2006b], and

are briefly presented below. To be able to simulate both sub-

and supercritical flow conditions, the backwater equation is

employed for low Froude number conditions and quasi-

6 of 16

W10436

CUI: TUGS MODEL

W10436

Page 7

normal flow assumption is applied for high Froude number

flow conditions.

dh

dx¼S0? Sf

1 ? F2

Sf¼ S0;

r

;

Fr? Frn

Fr> Frn

8

:

<

ð13Þ

in which h denotes water depth; x denotes downstream

distance, S0denotes local bed slope; Sf denotes friction

slope; and Frdenotes Froude number, which is calculated

by assuming a wide rectangular channel:

Fr¼

Qw

p

Bh

ffiffiffiffiffi

gh

ð14Þ

in which Qwdenotes water discharge; B denotes channel

width, which is generally chosen as bankfull width and

assumed to be a function of location but does not change in

time; and g denotes acceleration of gravity. Frn in

Equation (13) is the critical Froude number that separates

the application of backwater equation and quasi-normal

flow assumption. Here an arbitrary value of Frn= 0.8 is

used in the simulation, and any Frnvalue of between 0.75

and 0.9 produces similar results. The friction slope Sfis

calculated with the Keulegan formulation below,

Qw

p

Bh

ffiffiffiffiffiffiffiffiffiffi

ghSf

¼ 2:5ln 11h

ks

??

ð15aÞ

in which ksdenotes roughness height and is assumed to be

proportional to surface layer geometric mean grain size of

combined gravel and sand,

ks¼ 2:5Dsg¼ 2:5DFs

gsD1?Fs

sgg

ð15bÞ

in which Dsgdenotes the geometric mean grain size of the

surface layer forcombined gravel andsand; and Dsggdenotes

the surface gravel geometric mean grain size. Because ksis

located inside the log function in Equation (15a), a specific

choice of a ksvalue is not particularly sensitive to model

simulation. Here in Equation (15b) ksis assumed to be

2.5 times of the surface geometric mean grain size as it

produces reasonable results as demonstrated later in this

paper. Because the model is designed primarily for simula-

tion of gravel bedded rivers, and also for simplicity, form

friction is assumed to be relatively unimportant and is

ignored in the current model. Form friction may become

important once both surface and subsurface become

predominantly sandy, allowing for easier formation of bed

forms (e.g., dunes, anti-dunes).

[14] Combining the backwater equation and quasi-normal

flow assumption under different Froude number flow con-

ditions allows for a relatively simple algorithm in simulating

sub- and supercritical flow conditions, and the results from

the simulation are almost identical to more complicated

methods [e.g., Cui et al., 1996, 2005, and Cui and Parker,

1997] as demonstrated by Cui et al. [2005], and Cui et al.

[1996].

7.The Exner Equations of Sediment Continuity

[15] The Exner equations of sediment continuity are

modified from those by Parker [1991a, 1991b] to include

the presence of sand, and differ from those by Cui et al.

[2006a, 2006b] in that Cui et al. [2006a, 2006b] considered

gravel to form the frame of the deposit as the channel

aggrades, while sand only fills into the pores of the gravel

deposit. The current model considers both gravel and sand

as the matrix of the deposit. It is useful to point out that the

Exner equations of sediment continuity is explained only

briefly in this paper due to space limitations. The equations,

however, are similar to those by Parker [1991a, 1991b], Cui

and Parker [1998, 2005], and Cui et al. [2003b, 2006a,

2006b] in many ways. In particular, the equations presented

by Cui et al. [2006a, 2006b] are discussed in detail by

Stillwater Sciences [2002], which should provide adequate

information for interested readers to fully understand the

different components in the equations presented below. The

Exner equations for sediment continuity are:

1 ? lp

??B@h

@tþ@ Qgþ Qs

??

@x

þ 2 ? Fs

ðÞbaQg¼ qglþ qsl ð16aÞ

1 ? lp

??B

?

@x

baQg

3‘n 2 ð Þ

@ 1 ? Fs

ðÞLaFj

??

@t

þ 1 ? fIs

ðÞfIj@ h ? La

ðÞ

@t

??

þ

@ Qgpj

?

þ baQgpjþ 1 ? Fs

pjþ 1 ? Fs

yjþ1? yj

ðÞF0

j

hi

þ

ðÞF0

j

?pjþ1þ 1 ? Fs

yjþ2? yjþ1

ðÞF0

jþ1

!

¼ qglj

ð16bÞ

1 ? lp

þ@Qs

??B

@ FsLa

ð

@t

baQg

3‘n 2 ð Þ

Þ

þ fls@ h ? La

p1þ 1 ? Fs

y2? y1

ðÞ

@t

??

¼ qsl

@x?

ðÞF0

1

ð16cÞ

in which lpdenotes porosity of the sediment deposit; h

denotes the thickness of sediment deposit; t denotes time, ba

denotes volumetric abrasion coefficient (fraction of volume

lost per unit distance transported); qgland qsldenote lateral

gravel and sand input rates per unit distance (e.g., from bank

erosion or tributaries); Ladenotes surface layer thickness; pj

denotes fraction of the j-th size group of the gravel class of

the bed load (and p1is for j = 1); Fj0is an areal estimate of

the fraction of exposure of surface gravel of the j-th group

(and F1

0is for j = 1) given by Parker [1991a, 1991b] below:

F0

j¼

Fj=

ffiffiffiffiffi

q

Dj

q

X

Fj=

ffiffiffiffiffi

Dj

??

ð17Þ

qgljdenotes the lateral gravel input rate per unit distance for

the j-th size group, so that Sqglj= qgl; and yjdenotes base-2

logarithmic grain size psi-scale associated with grain size

Dj, i.e.,

yj¼ log2Dj

??;

j ¼ 1;2;...;N;N þ 1

ð18Þ

Equation (16a) is the conservation of total sediment load

(sand and gravel), in which the last term on the left hand

W10436

CUI: TUGS MODEL

7 of 16

W10436

Page 8

side is a sink term, representing gravel loss to silt as gravel

abrades into smaller gravel particles, sand, and silt [Parker,

1991a, 1991b]. No sink term for sand is present in

Equation (16a) because the equation considers the mass

conservation for combined gravel and sand. Equation (16b)

is the conservation of gravel in the j-th size group, in which

the first term involving barepresents gravel mass lost to silt

by abrasion and the second term involving barepresents

gravel mass transfer between adjacent size groups through

abrasion. The factor (1 ? Fs) in the abrasion term is the

result that only (1 ? Fs) of the surface layer is composed of

gravel, and the factor (2 ? Fs) in Equation (16a) is the result

of S[pj+ (1 ? Fs)Fj0]. Equation (16c) is the conservation of

sand, in which the last term on the left hand side represents

increased sand mass through abrasion of gravel, which is

also seen in Equation (16b) for the finest gravel size group

(i.e., for j = 1).

[16] For simplicity, the hypothetical sand entrainment and

infiltration relations are not included in the Exner equations

of sediment continuity (Equations (16a), (16b), and (16c)).

Instead, a relatively simple procedure is used to re-update

the subsurface sand fraction and adjust sand transport rate

following the solution of the Exner equations above, once

sand entrainment or infiltration occurs. At each step, the

subsurface sand fraction is updated as

fs¼ f0

s?

qvsDt

1 ? lp

??Hei

ð19Þ

in which fs0denotes the subsurface sand fraction calculated

without considering sand entrainment and infiltration; Dt

denotes time increment; and Heidenote the depth of the

subsurface that subjects to entrainment and infiltration. The

addition or reduction of sand transport rate through sand

entrainment or infiltration is then factored into the overall

mass balance with:

qsl¼ q0

slþ qsvB

ð20Þ

in which qsl0denotes lateral sediment input, shown in

Equations (16a) and (16c) as qsl. In addition to the

adjustment of fs and qsl due to sand entrainment and

infiltration, porosity lpis adjusted by replacing the entrained

sand with porous space for entrainment and replacing the

porous space with sand for infiltration. According to this, the

following formulation can be used to adjust the porosity

whenever subsurface sand entrainment or infiltration occurs:

fsþ

lp

1 ? lp¼ f0

sþ

l0

p

1 ? l0

p

ð21Þ

in which lp0denotes porosity before adjustment and lp

denotes porosity after adjustment.

8. Brief Description of Solution Procedure

[17] The sediment transport equation of Wilcock and

Crowe [2003] (Equations (1) through (5)), gravel transfer

functions (Equations (6a) and (7)), sand transfer functions

(Equations (6b) and (8)), sand entrainment and infiltration

functions (Equations (9), (10), (11a), (11b), and (12)),

governing equation for flow (Equation (13)), friction for-

mulation (Equation (15a)), and the Exner equations of

sediment continuity (Equations (16a), (16b), (16c), (19),

and (20)) are organized and solved with FORTRAN pro-

gram language to form The Unified Gravel-Sand (TUGS)

Model. A brief procedure for the solution is provided below.

[18] Note the partial differential equations (PDE) are

discretized as first order accurate both in space and in time;

the discretization in time is explicit, and the spatial discre-

tization is conducted with the standard upwind scheme [e.g.,

Hirsch, 1989]. The PDEs indicated in the following brief

description of solution procedure all refers to their discre-

tized form.

[19] . Solve the backwater equations and resistance

(friction) relations, Equations (13), (14), (15a), and (15b)

based on the information at time t;

[20] . Calculate gravel and sand transport rates and bed

load grain size distribution with Wilcock and Crowe [2003]

equation, Equations (1), (2a), (2b), (3a), (3b), (3c), (4), and

(5a), (5b), (5c);

[21] . Solve Equation (16a) to determine the amount of

aggradation or degradation between time t and t + Dt;

[22] . Calculate interface sediment grain size distribution

as represented with parameters fIs, fI1, fI2, ..., fIN with

Equations (6a), (6b), or (7) and (8), depending on whether

the bed is degrading or aggrading;

[23] . Solve Equations (16b) and (16c) to update surface

grain size distribution as represented by parameters Fs, F1,

F2, ..., FN;

[24] . In case of aggradation, update subsurface sediment

grain size distribution by mixing the interface sediment with

the top layer of the subsurface sediment (details not pre-

sented but the mixing is a linear combination and is very

simple to implement);

[25] . Adjust fs, qsl, and lpfor sand entrainment and

infiltration with Equations (9), (10), (11a), (11b), (12), (19),

(20), and (21);

[26] . Update all the record for time t + Dt and go to the

next time step.

9.

Runs

[27] Model performance is examined with three SAFL

downstream fining narrow flume runs [Paola et al., 1992;

Seal et al., 1995, 1997; Toro-Escobar et al., 1996]. The

three experimental runs were conducted in a 0.305-m wide

and 50-m long flume with an initial concrete-bottom slope

of 0.002 (Figure 3). The grain size distribution of the

sediment used to feed the flume is shown in Figure 4,

which is 33.1% sand and 66.9% gravel, and has a geometric

mean grain size of 4.63 mm and a geometric standard

deviation of 5.57 [Cui et al., 1996]. Each of the three runs

is conducted with a constant water discharge and a constant

sediment feed rate. The flume is ponded at its downstream

reach by setting a constant water surface elevation at the

downstream end, which drives channel aggradation and

downstream fining. The relevant parameters for the three

runs are given in Table 1. During the experiments, channel

bed elevations were monitored at several intervals. Samples

of the sediment deposits were then collected and analyzed

following the termination of each run. The samples were

taken in layers and were labeled with the duration the layer

of sediment was deposited (e.g., 5–10 h denotes the layer of

Simulating SAFL Downstream Fining Flume

8 of 16

W10436

CUI: TUGS MODEL

W10436

Page 9

sediment that was deposited between 5 and 10 h). Full

description of the experimental data is given by Seal et al.

[1995].

[28] Three previous numerical simulations had been con-

ducted by this author and his co-contributors to simulate the

three SAFL downstream fining experiments [i.e., Cui et al.,

1996, 2006b; Cui and Parker, 1997]. Parker’s surface-based

bed load equation had been used in all these three previous

efforts. Among other things, the major differences in the

three previous simulations are in the methods used to solve

the flow parameters. In the simulation of Cui et al. [1996], a

time-relaxation method was used to solve the non-steady

flow equations in order to reach a steady flow solution for

the transient flow, which is a rather inefficient solution by

today’s standards. In the simulation of Cui and Parker

[1997], a shock-fitting method was used to capture the

precise position of the sediment wedge front and hydraulic

jump so that a quasi-normal flow assumption could be

applied upstream of the hydraulic jump while a simple

backwater calculation was used downstream of the hydrau-

lic jump. Because of the good results of Cui and Parker

[1997], Cui et al. [2003a] and Cui and Parker [2005]

simplified the procedure and started to calculate flow

parameters with a combination of quasi-normal flow as-

sumption for high Froude number reaches and simple

backwater calculation for low Froude number reaches, and

Cui et al. [2006b] simulated the SAFL downstream fining

Run 2 with this simplified method to demonstrate that it

produced almost identical results compared with that in Cui

et al. [1996] and Cui and Parker [1997]. This simplified

method is also used in TUGS model presented in this paper

and by Cui and Wilcox [2007]. Because Parker’s [1990]

equation excludes sand from the simulation, the previous

efforts of Cui et al. [1996, 2006b] and Cui and Parker

[1997] for the SAFL downstream fining runs did not

simulate fraction of sand in the deposit, whereas the TUGS

simulation presented below simulates the fraction of sand in

the deposit in addition to the previously simulated bed

profile and gravel characteristic grain sizes, and compares

the results with experimental observations.

9.1. TUGS Model Simulation With Unmodified

Wilcock and Crowe [2003] Equation

[29] Because the SAFL experimental runs applied con-

stant water discharge and sediment feed, the gravel and sand

are co-deposited gradually onto the flume bed and there is

no additional sand infiltration and entrainment. With that,

parameter aiand aewere both set to zero for the numerical

simulation (and because of this, the value of Heiis irrelevant

to this simulation, and porosity lpbecomes a constant). The

porosity lp is set to a constant value of 0.3 for the

simulation, which is a typical value for deposits of gravel

sand mixtures [e.g., Wu and Wang, 2006]. Because the

experiment is conducted in a flume with a length of less

than 50 m, particle abrasion is negligible, and thus, abrasion

coefficient bais set to zero. Also, because there was no

lateral input of sediment, parameters qgland qslare set to

zero. Active layer thickness is set to a constant value of

5 cm, which is slightly larger than the D90(i.e., the grain size

that 90% of the sediment is finer than) of the feed sediment.

Cui et al. [2006a] had demonstrated that the selection of

Figure 3. Flume set up for SAFL downstream fining experiments.

Figure 4. Grain size distribution of the sediment used for

sediment feed in the three narrow runs of SAFL down-

stream fining experiments.

Table 1. Relevant Parameters in SAFL Downstream Fining

Experiments, Narrow Runs

Run123

Water discharge, l/s

Sediment feed rate, kg/min

Adjusted sediment feed rate,akg/min

Experimental duration, h

Downstream end water surface elevation, m

49 49

5.65

5.33

32.4

0.45

49

2.83

2.78

64

0.50

11.30

11.24

16.83

0.40

aSediment feed rates were adjusted to account for the sediment that rolled

backward following sediment feed, as illustrated in Figure 3 [Cui et al.,

1996].

W10436

CUI: TUGS MODEL

9 of 16

W10436

Page 10

active layer thickness is not particularly sensitive to mod-

eling results, as long as it is within a reasonable range. The

first attempt is made to simulate the three flume runs with

the unadjusted TUGS model, i.e., no adjustment is made to

the coefficients in Wilcock and Crowe’s [2003] equation.

Model results are shown in Figures 5a and 5b for bed

elevations for Runs 1 and 3, respectively, in Figure 5c for

gravel characteristic grain sizes in the deposits for Run 3,

and in Figure 5d for sand fractions in the deposits for Run 3.

Results in Figure 5 indicate that TUGS model, without any

modification to Wilcock and Crowe’s [2003] equation,

excellently reproduced the grain size distributions of the

sediment deposit, as indicated in the comparison of gravel

characteristic grain sizes and sand fractions in the deposit

(Figures 5c and 5d). The simulated gravel characteristic

grain sizes (y10, y50, and y90) shown in Figure 5c, for

example, (a) closely match the observed values; (b) decrease

in the downstream direction similar to observed in the

flume; and (c) gradually increase in time to approach

their equilibrium values (i.e., the values at x = 0). The

simulated sand fractions shown in Figure 5d also closely

match the observed values and the trend both in space and

in time with the exception that there are two exceptionally

high sand fraction values for the time interval between 6 and

14 h in the experiment that is not reproduced in the

simulation. The observed two high sand fraction values

are samples from a small amount of sand deposit down-

stream of the main depositional front, as indicated in

Figures 5b and 5d. During the experiment, as the majority

of the gravel and sand is deposited as a sediment wage that

gradually build upward with its front gradually migrated

downstream, a small amount of sand transported passed the

depositional front, forming a small amount of sand deposit

that were later buried by the main deposit as the front

migrated downstream. This small amount of sand that

passed the depositional front is not produced in this simu-

lation. Simulation results for bed elevation indicate that the

model over-predicted bed slope for all the three runs, as

shown for Runs 1 and 3 in Figures 5a and 5b, and the

simulated slope is more accurate as sediment feed rate

decreases (i.e., Run 3 is better than Run 2, and Run 2 is

better than Run 1). This observation indicates that some

minor adjustments to the Wilcock and Crowe [2003] equa-

tion that increase its predicted sediment transport rate,

particularly when shear stress is high, should allow for a

better prediction of bed slope. The simulated volume of

sediment deposit seems to be higher than the experimental

data as shown in Figures 5a and 5b. One of the concerns is

that a programming error had resulted in over predicted

sediment mass in the simulation. To make sure that the

numerical model conserves mass, a hand check of deposi-

Figure 5. Simulated results with TUGS model, unmodified Wilcock and Crowe [2003] equation, in

comparison with observations from flume measurements: (a) bed elevation for Run 1; (b) bed elevation

for Run 3; (c) gravel characteristic grain sizes for Run 3, and (d) sand fraction in the deposit for Run 3. In

(c), the lines are predicted results, alternating between solid and dashed lines at different times for better

visualization.

10 of 16

W10436

CUI: TUGS MODEL

W10436

Page 11

tionalvolumeisconductedanditwasconfirmedthat

Zt

0

(Qg+

Qs)dt is almost identical to [(1 ? lp)

ZL

0

hdx]at time t, where L

denotes flume length. That is, the numerical simulation

satisfies mass conservation. Another possibility is that the

porosity value used in the simulation, lp= 0.3, is higher

than the porosity of the sediment deposit in the flume. The

newly deposited sediment with a grain size distribution used

in the experiment, however, rarely has a porosity value

below 0.3, as indicated by the equations of Komura [1963]

and Han et al. [1981], and both equations can be found in

Wu and Wang [2006]. With that, the porosity value should

not be arbitrarily reduced to below 0.3 without any porosity

measurements during the experiment. The most likely

reason for the smaller sediment deposit volume than the

simulation is that some fine sediment passed the deposi-

tional front and cannot be counted for in the bed profile

comparison. In addition, potential deviation from the

designed sediment feed during the flume experiments is

also a possibility because sediment of the size (up to 90 mm

in diameter) used in the experiments cannot be fed with a

relatively accurate sediment feeder. The implication of the

over predicted volume of sediment deposition will be

discussed later following a revised simulation.

9.2. TUGS Model Simulation With Slightly Modified

Wilcock and Crowe (2003) Equation

[30] Because the numerical model over predicted the bed

slope, and the discrepancy between numerical simulation

and flume experiment increases with the increase in sedi-

ment transport rate, an increase in the calculated sediment

transport rate and an increase in the slope of the Wj*? t/trj

relation (Equation (1)) of the Wilcock and Crowe [2003]

equation will improve the numerical prediction. Here the

sediment transport data collected by Wilcock and Crowe

[2003] and used to derive their bed load equation are plotted

in Figure 6, along with their Wj*? t/trjrelation (the data

are downloaded from ftp://agu.org under subdirectory

2001WR000683 in June 2006). In addition, the Wj*? t/trj

relation of Parker [1990] is also plotted in the diagram for

comparison purposes. For those who are familiar with

Wilcock and Crowe [2003] equation, the plotting positions

of the experimental data (shown in Figure 6 as symbols) are

slightly different from those presented in Wilcock and

Crowe [2003] because, for consistency, this analysis applied

the resistance equations presented in this paper to calculate

shear stress instead of applying the normal flow assumption

used in the original analysis.

[31] It can be observed in Figure 6 that the Wj*? t/trj

relation by Parker [1990] is steeper than that of Wilcock and

Crowe [2003], and the Wj*values calculated with Parker

[1990] relation is generally higher than those calculated

with Wilcock and Crowe [2003] relation for higher t/trj

values. As a result, a trial TUGS model run is conducted by

replacing the original Wilcock and Crowe [2003] Wi*? t/tri

relation with that of Parker [1990]. To be consistent with

Wilcock and Crowe’s [2003] equation, dimensionless sedi-

ment transport rate for the Parker [1990] Wj*? t/trj

relation is adjusted to 0.002 from its original value of

0.00218, which is a relatively minor adjustment:

W*

j¼

10:95 1 ?0:853

0:002exp 14:2 t=trj? 1

0:002 t=trj

?

t=trj

??4:5

;

for t=trj> 1:59

??? 9:28 t=trj? 1

??2

hi

;

for 1 < t=trj? 1:59

for t=trj? 1

?14:2;

8

>

>

>

>

>

>

>

>

<

:

ð22Þ

SimulatedresultsbyreplacingEquation(1)withEquation(22)

are presented in Figures 7, 8, and 9, for Runs 1, 2, and 3,

respectively. Comparison of profiles in Figures 7a, 8a,

and 9a shows that numerical simulation adequately repro-

duced the observed bed profile as indicated by the very close

bed slope between numerical simulation and observation,

Figure 6.

normalized shear stress relation. Experimental data are

analyzed by applying Manning-Strickler resistance equation

and water continuity equation to calculate water depth and

shear stress. Data used for analysis come from the original

data set of Wilcock and Crowe [2003] downloaded from

ftp://agu.org under subdirectory 2001WR000683 in June

2006. The dimensionless sediment transport rate ? normal-

ized shear stress relations of both Parker’s [1990] and

Wilcock and Crowe’s [2003] are presented with the data.

Dimensionless sediment transport rate ?

W10436

CUI: TUGS MODEL

11 of 16

W10436

Page 12

improving significantly from the original simulation results

as presented in Figures 5a and 5b. The simulated volume of

sediment upstream of the depositional front remains to be

higher than the observed value for all the three runs, most

likely due to the fact that some fine sediment bypassed the

depositional front in the flume experiment as discussed

earlier. The model is unable to bypass the small amount of

fine sediment through the depositional front, which is likely

an area for future improvement. Comparison between

Figures 9b and 5d indicates that the simulated sand fractions

in the deposit are slightly worse than the original simulation.

Despite this slightly decreased simulation quality, the

simulated sand fraction adequately matches those observed

in the experiments as shown in Figures 7b, 8b, and 9b.

Comparisons between simulation and flume experiment in

Figures 7c, 8c, and 9c indicate that the adjusted model

Figure 7. Simulated (a) bed profile, (b) sand fraction; and

(c) characteristic gravel grain size in the deposit for Run 1

by replacing Wilcock and Crowe’s dimensionless sediment

transport rate ? normalized shear stress relation with that of

Parker [1990]. In (c), the lines are predicted results,

alternating between solid and dashed lines at different times

for better visualization.

Figure 8. Simulated (a) bed profile, (b) sand fraction; and

(c) characteristic gravel grain size in the deposit for Run 2

by replacing Wilcock and Crowe’s dimensionless sediment

transport rate ? normalized shear stress relation with that of

Parker [1990]. In (c), the lines are predicted results,

alternating between solid and dashed lines at different times

for better visualization.

12 of 16

W10436

CUI: TUGS MODEL

W10436

Page 13

adequately reproduced subsurface gravel characteristic grain

sizes, similar to the original model.

9.3. Discussions on the SAFL Downstream Fining

Simulation

[32] Comparison between simulated and observed sedi-

mentation process for the three SAFL downstream fining

runs indicated that TUGS model, with slight modification to

the equation of Wilcock and Crowe’s [2003], adequately

reproduced general bed slope, the subsurface gravel charac-

teristic grain sizes, and the fraction of sand in the deposit. The

area where the model needs improvement is that numerical

simulation over predicted the volume of sediment deposition

upstream of the depositional front due to the fact that the

model was unable to bypass the small amount of fine sand

through the depositional front to the impoundment area. In

other word, the model under predicted the transport rate for

the finest fraction of the sand class when shear stress is low

(i.e., in the ponded area). Even without further improvement,

the model should perform reasonably well in simulating

sediment transport under most circumstances judged by the

goodagreementbetweenthesimulationandflumeexperiment

in bed slope, gravel grain size, and sand fractions in the

deposit. If we consider the under prediction of the finest

fraction of the sand class under low shear stress as a

systematic error in the predicted sediment transport rate, this

error is no more than 10% of the overall sediment transport

rate for the cases simulated, judged by the no more than 10%

of over deposition of the sediment volume upstream of the

depositional front. In all the practical problems, the relative

error in sediment supply, which serves as model input, is

usually much larger than 10%, making a 10% relative error in

predicted transport rate acceptable. This argument can also be

corroborated by the sediment transport ? shear stress data

such as shown in Figure 6, where both Wj*and t/trjare

plotted in log-scale, and for any given t/trjvalue, Wj*varies

by more than an order of magnitude.

10.

Flushing Flow Flume Experiment of Wu and Chou

[2003]

[33] Wu and Chou [2003] conducted a flushing flow

experiment with a 40-cm wide flume. The flume is 7.2-m

long with a 2.5-m long section in the middle designated as

the experimental section. The flume is set at a slope of 0.01,

a typical value for gravel bedded rivers. A gravel mix with

grain size ranging between 2 and 50.8 mm and a sand mix

with grain size ranging between 0.5 and 2 mm are used for

the experiment. In the experimental section, a mixture of

32% sand mix and 68% gravel mix was placed in the flume

with a thickness of 10 cm. The reaches upstream and

downstream of the experimental section were placed with

10-cm thick gravel mix without sand. In order to observe

the effect of flushing flow, a constant discharge of 68 l/s was

sent through the flume for 7 h. To sample fine sediment

fractions in the deposit through time, the experimental

section was divided into three reaches, each occupies a

third of the 2.5-m experimental zone. The three reaches

were named Reaches 1, 2, and 3 from upstream to down-

stream. Bed material was sampled by inserting caped 9-cm

diameter 6-cm long cylinders into the deposit at different

times during the run and leaving them until the end of the

run. The cylinders had caps that prevent further fine

sediment flushing and preserved the samples for grain size

analysis at the end of the experiment. Upon finishing the

experiment, the samples within the cylinders were divided

into two 3-cm subsamples, with the upper sample termed as

surface and bottom sample termed as subsurface. The

surface and subsurface samples were analyzed separately

Examination of Sand Entrainment With

Figure 9. Simulated (a) bed profile, (b) sand fraction; and

(c) characteristic gravel grain size in the deposit for Run 3

by replacing Wilcock and Crowe’s dimensionless sediment

transport rate ? normalized shear stress relation with that of

Parker [1990]. In (c), the lines are predicted results,

alternating between solid and dashed lines at different times

for better visualization.

W10436

CUI: TUGS MODEL

13 of 16

W10436

Page 14

to obtain sand fraction values. During the experiment, 3 kg

of gravel was added to the upstream end of the flume as a

slug in every 30 min. More details of the experiment can be

found in the original reference [Wu and Chou, 2003].

[34] Simulation of the Wu and Chou [2003] flushing flow

experiment with TUGS model followed exactly the same

procedure as the flume experiment described above. Similar

to the simulation of the SAFL experiments, abrasion coef-

ficient bawas set to zero due to the limited channel length.

Because there is no fine sediment infiltration within the

experimental zone, coefficient aiin Equation (11b) is set to 0.

Similar to the simulation of SAFL downstream fining runs,

initial porosity of the gravel and sand deposit is set at 0.3

and both the surface layer thickness and parameter Heiin

Equation (19) are set to 3 cm to be consistent with Wu and

Chou’s [2003] sampling protocol. Because there was no

lateral sediment input during the experiment, both qgland

qslare initially set to zero for the simulation. Note that while

qglremains to be zero throughout the simulation, qslwill

become higher than zero during simulation because the

entrained sediment during flushing flow is wrapped into

this term with the application of Equation (20).

[35] Several trials are needed for adjusting coefficient aein

Equation (11a) in order to achieve a reasonable fit between

the flume experiment and numerical simulation. The results

presented below used the unmodified Wilcock and Crowe

[2003] equation and a coefficient aevalue of 0.02.

[36] Comparison of simulated and observed results is

presented in Figure 10, which indicates that numerical

simulation reasonably reproduced changes in bed elevation,

subsurface sand fraction and sand fraction in combined

surface/subsurface sample, while the simulated surface layer

has less fine sediment than the sampling indicated. The

original publication of Wu and Chou [2003] compared the

change in bed elevation, surface sand fraction, and subsur-

face sand fraction. Here I added the comparison of sand

fraction in the combined surface and subsurface sample.

Because of the way the samples were separated into the

surface and subsurfacelayers, thesand fractions incombined

surface and subsurface samples should be more reliable than

the data for surface and subsurface samples. The photograph

showing channel bed before and after flushing flow by Wu

and Chou [2003], for example, indicated that the post-

flushing channel surface for Reach 2 is relatively free of

sand, indicating that the way to separate the 6-cm samples

into surface and subsurface sub-samples may have some-

what influenced the surface sand fraction data.

[37] Note that the formulation for sediment infiltration

(Equation (11b)) is not tested in this paper. It is presented here

as a compliment to the entrainment formulation because

infiltration can be important in short-term and can act to negate

the entrainment at long-term basis. Both Equations (11a)

and (11b), although somewhat useful in providing fine

sediment infiltration and entrainment information if cali-

Figure 10. Comparison of simulated and observed results for the Wu and Chou [2003] flushing flow

experiment: (a) changes in bed elevation; (b) surface layer sand fraction; (c) subsurface sand fraction; and

(d) sand fraction in the combined surface and subsurface sample. The predicted results are solid lines and

the measured results are symbols connected with dashed lines.

14 of 16

W10436

CUI: TUGS MODEL

W10436

Page 15

brated to the specific river reach, should be replaced with

more physically based equations that need minimal or no

calibration, once such equations are available. For long-term

simulations under natural conditions, it can be assumed that

the entrainment and infiltration of sand achieve a dynamic

balance, and thus, can be neglected in the simulation by

setting both aiand aeto zero such as in the simulation of the

SAFL downstream fining experiments. In a paper submitted

concomitantly to River Research and Applications [Cui,

2007], I examine the dynamics of sand fractions in Sandy

River, Oregon, including the sedimentation process up-

stream of Marmot Dam and a 50-km reach between Marmot

Dam and Columbia River Confluence under the assump-

tion that ai= 0 and ae= 0.

11. Conclusions

[38] The Unified Gravel-Sand (TUGS) model is devel-

oped based on Wilcock and Crowe’s [2003] bed load

equation, the gravel transfer function of Hoey and Ferguson

[1994] and Toro-Escobar et al. [1996], a hypothetical sand

transfer function during bed aggradation, and hypothetical

subsurface sand entrainment/infiltration functions. The

model, without adjustment to any coefficient in the Wilcock

and Crowe’s [2003] equation, are applied to simulate three

relatively large-scale flume experiments and produced ex-

cellent agreements between simulated and measured char-

acteristic gravel grain sizes and fraction of sand in the

deposits. The unmodified model, however, over-predicted

the bed slope for all the runs, and a minor adjustment to the

model produced good agreements in bed profiles, gravel

characteristic grain sizes and sand fractions in the deposits

for all the three runs. To test the performance of sand

entrainment from subsurface deposit, the model is tested

against the flushing flow data of Wu and Chou [2003] with

reasonable results. This reasonable agreement, nevertheless,

should not be interpreted as an indication that the model can

simulate sand entrainment accurately for other situations.

What it suggests is that, the model can be used for

simulation of fine sediment entrainment if adequate calibra-

tion is conducted. More physically based fine sediment

entrainment and infiltration functions are needed so that

the model can be applied in rivers with minimal or no

calibration in simulating short-term events such as fine

sediment entrainment during and fine sediment infiltration

after the release of a flushing flow.

Notations

ae, ai

coefficients in entrainment and infiltration functions;

channel width;

particle diameter separating the (j-1)-th and j-th size

groups;

mean particle diameter of the j-th size group,

Dj¼

geometric mean grain size of sand;

surface geometric mean grain size for the combined

sand and gravel;

surface geometric mean grain size of the gravel;

fraction of j-th size group of the surface layer gravel;

aerial fraction of the j-th size group of surface layer

gravel;

Froude number;

B

Dj

Dj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

DjDjþ1

p

;

Dgs

Dsg

Dsgg

Fj

Fj0

Fr

Frn

critical Froude number that defines whether to apply

the backwater equation or the quasi-normal flow

assumption;

surface layer sand fraction;

equilibrium surface sand fraction for sand entrain-

ment;

fraction of the j-th size group of the gravel class for

sediment that transfers from the bed load and surface

layer to the subsurface;

sand fraction of the sediment that transfers from the

bed load and surface layer to the subsurface;

sand fraction within subsurface sediment deposit;

equilibrium subsurface sand fraction for sand

infiltration;

acceleration of gravity;

water depth;

roughness height;

surface layer thickness;

fraction of the j-th size group of the gravel class in

bed load;

volumetric transport rate of gravel;

volumetric transport rate of sand;

water discharge;

upward sand flux per unit area from subsurface sand

entrainment;

lateral volumetric gravel supply rate per unit channel

length;

lateral volumetric gravel supply rate of the j-th size

group per unit channel length;

downward sand flux per unit area from sand

infiltration;

lateral volumetric sand supply rate per unit channel

length;

net upward flux from combined sand entrainment

and infiltration;

bed slope or water surface slope;

local bed slope;

local friction slope;

time;

settling velocity of fine sediment particles;

dimensionless sediment transport rate;

volumetric abrasion coefficient of gravel;

coefficient in gravel transfer function;

thickness of the sediment deposit;

porosity of the sediment deposit;

density of water;

geometric standard deviation of subsurface gravel;

geometric standard deviation of surface layer gravel;

shear stress;

reference shear stress;

reference shear stress for surface geometric mean

grain size;

grain size psi-scale associated with grain size

Dj, yj= log2(Dj).

Fs

Fse

fIj

fIs

fs

fse

g

h

ks

La

pj

Qg

Qs

Qw

qe

qgl

qglj

qi

qsl

qsv

S

S0

Sf

t

vs

Wi*

ba

c

h

lp

r

sgg

ssgg

t

tri

trm

yj

[39] Acknowledgments.

by CALFED Ecosystem Restoration Program (Grant ERP-02D-P61) and

The Nature Conservancy (TNC). I thank Keith Barnard, Matt Brown, Geoff

Hales, Graham Matthews, Scott McBain, Jess Newton and John Wooster

for providing bulk sampling data, Rebecca Soileau for providing her

experimental data, and Peter Wilcock for making his experimental data

available on the Web. The assistance, comments and critiques from

Christian Braudrick, Bill Dietrich, Mike Fainter, Mike Roberts, Jeremy

Venditti and John Wooster are gratefully acknowledged. I would also like to

Funding for model development is provided

W10436

CUI: TUGS MODEL

15 of 16

W10436

Page 16

thank Stillwater Sciences for providing financial support during the draft

and revisions of this manuscript. The useful comments from Rob Ferguson,

Marwan Hassan, two anonymous reviewers, associate editors (Andre ´ Roy

and Tammo Steenhuis) and the editor (Scott Tyler) have been incorporated

into the manuscript.

References

Beacham, T. D., and C. B. Murray (1990), Temperature, Egg Size, and

Development of Embryos and Alevins of Five Species of Pacific Salmon:

A Comparative Analysis, Transactions of the American Fisheries Society,

vol. 119, 927–945.

Beschta, R. L., and W. L. Jackson (1979), The intrusion of fine sediments

into a stable gravel bed, J. Fish. Res. Board Ca., 36, 204–210.

CDWR (California Dept. of Water Resources) (1994), San Joaquin River

tributaries spawning gravel assessment, Stanislaus, Tuolumne, and

Merced Rivers, Appendix C: Bulk sampling data, surface and subsurface,

September.

CDWR (California Dept. of Water Resources) (1995), Sacramento River

gravel study – Keswick Dam to Cottonwood Creek, Memorandum to

Stacy Cepello, Environmental Specialist IV, and Koll Buer, Senior

Engineering Geologist, October 20.

Coble, D. W. (1961), Influence of water exchange and dissolved oxygen in

redds on survival of steelhead trout embryos, Trans. Am. Fish. Soc., 90,

469–474.

Cooper, A. C. (1965), The effects of transported stream sediments on the

survival of sockeye and pink salmon eggs and alevin, Bulletin 18, Inter-

national Pacific Salmon Fisheries Commission, New Westminster, Brit-

ish Columbia, Canada.

Cui, Y. (2007), Examining the dynamics of grain size distributions of

gravel/sand deposits in the Sandy River, Oregon with a numerical model,

River Research and Applications, 23, 732–751, doi:10.1002/rra.1012.

Cui, Y., and G. Parker (1997), A quasi-normal simulation of aggradation and

downstream fining with shock fitting, Int. J. Sediment Res., 12(2), 68–82.

Cui, Y., and G. Parker (1998), The arrested gravel-front: stable gravel-sand

transitions in rivers. Part 2: General numerical solution, J. Hydraul. Res.,

36(2), 159–182.

Cui, Y., and G. Parker (2005), Numerical model of sediment pulses and

sediment supply disturbances in mountain rivers, J. Hydraul. Eng.,

131(8), 646–656, doi:10.1061/ (ASCE)0733-9429 (2005)131:8 (646).

Cui, Y., and A. C. Wilcox (2007), Numerical modeling of sediment trans-

port upon dam removal: application to Marmot Dam in Sandy River,

Oregon, Chapter 23 in Sedimentation Engineering, ASCE Manual 110,

M. H. Garcia, in press.

Cui, Y., G. Parker, and C. Paola (1996), Numerical simulation of aggrada-

tion and downstream fining, J. Hydraul. Res., 32(2), 185–204.

Cui, Y., G. Parker, T. E. Lisle, J. Gott, M. E. Hansler-Ball, J. E. Pizzuto,

N.E.Allmendinger,andJ.M. Reed(2003a),Sedimentpulsesin mountain

rivers, Part I: Experiment, Water Resour. Res., 39(9), 1239, doi:10.1029/

2002WR001803.

Cui, Y., G. Parker, J. E. Pizzuto, and T. E. Lisle (2003b), Sediment pulses in

mountain rivers, Part II: Comparison between experiments and numerical

predictions, Water Resour. Res., 39(9), 1240, doi:10.1029/2002WR001805.

Cui, Y., G. Parker, T. E. Lisle, J. E. Pizzuto, and A. M. Dodd (2005), More

on the evolution of bed material waves in alluvial rivers, Earth Surf.

Proc. Landforms, 30, 107–114, doi:10.1002/esp.1156.

Cui, Y., C. Braudrick, W. E. Dietrich, B. Cluer, and G. Parker (2006a), Dam

Removal Express Assessment Models (DREAM), Part 2: Sensitivity

tests/sample runs, J. Hydraul. Res., 44(3), 308–323.

Cui, Y., G. Parker, C. Braudrick, W. E. Dietrich, and B. Cluer (2006b), Dam

Removal Express Assessment Models (DREAM), Part 1: Model devel-

opment and validation, J. Hydraul. Res., 44(3), 291–307.

Diplas, P., and G. Parker (1985), Pollution of gravel spawning grounds due

to fine sediment, Project Report No. 240, St. Anthony Falls Laboratory,

Univ. of Minnesota, Minneapolis, Minnesota, 131 pp.

Ferguson, R. I. (2003), Emergence of abrupt gravel-to-sand transitions

along rivers through sorting processes, Geology, 31(2), 159–162,

doi:10.1130/0091-7613 (2003)031<0159:EOAGTS>2.0.CO;2.

Graham Matthews and Associates (2001), Gravel quality monitoring in the

mainstem Trinity River, Final Report prepared for Trinity County Board

of Supervisors, January, 21 p + figures and appendices.

Graham Matthews and Associates (2003a), WY2003 geomorphic monitor-

ing report, Report to Western Shasta Resource Conservation District,

Anderson, CA.

Graham Matthews and Associates (2003b), Hydrology, geomorphology,

and historic channel changes of Lower Cottonwood Creek, Shasta and

Tehama Counties, California, Report to National Fish and Wildlife Foun-

dation, CALFED Bay Delta Program Project # 97-N07.

Graham Matthews and Associates (2004), WY2004 geomorphic monitor-

ing report, Report to Western Shasta Resource Conservation District,

Anderson, CA.

Han, Q. W., Y. C. Wang, and X. L. Xiang (1981), Initial dry density of

sediment deposit, J. Sediment Research, Issue 1, (in Chinese).

Hirano, M. (1971), River bed degradation with armoring, Proc. Jpn. Soc.

Civil Eng., 195, 55–65.

Hirsch, C. (1989), Numerical Computation of Internal and External Flows,

Volume 1: Fundamentals of Numerical Discretization, John Wiley &

Sons, New Ed Edition, 538 p, ISBN-10:04719238501.

Hoey, T. B., and R. I. Ferguson (1994), Numerical simulation of down-

stream fining by selective transport in gravel bed rivers: Model develop-

ment and illustration, Water Resour. Res., 30, 2251–2260.

Komura, S. (1963), Discussion of ‘‘sediment transport mechanics: Intro-

duction and properties of sediment,’’ J. Hydraul. Division, ASCE, 89(1),

263–266.

Paola, C., G. Parker, R. Seal, S. K. Sinha, J. B. Southard, and P. R. Wilcock

(1992), Downstream fining by selective deposition in a laboratory flume,

Science, 258, 1757–1760.

Parker, G. (1990), Surface-based bedload transport relation for gravel riv-

ers, J. Hydraul. Res., 28(4), 417–436.

Parker, G. (1991a), Selective sorting and abrasion of river gravel, I: Theory,

J. Hydraul. Eng., 117(2), 131–149.

Parker, G. (1991b), Selective sorting and abrasion of river gravel, II:

Application, J. Hydraul. Eng., 117(2), 150–171.

Parker, G., and A. J. Sutherland (1990), Fluvial armor, J. Hydraul. Res.,

28(5), 529–544.

Phillips, R. W., R. L. Lantz, E. W. Claire, and J. R. Moring (1975), Some

effects of gravel mixtures on emergence of coho salmon and steelhead

trout fry, Trans. Am. Fish. Soc., 104, 461–466.

Ribberink, J. (1987), Mathematical modeling of one-dimensional morpho-

logical changes in rivers with non-uniform sediment, Ph.D.Thesis, Delft

Univ. of Technology, the Netherlands.

Seal, R., C. Paola, G. Parker, and B. Mullenbach (1995), Laboratory experi-

ments on downstream fining of gravel, narrow channel runs 1 through 3:

supplementalmethodsanddata,ExternalMemorandumM-239,St.Anthony

Falls Laboratory, Univ. of Minnesota.

Seal, R., C. Paola, G. Parker, J. B. Southard, and P. R. Wilcock (1997),

Experiments on downstream fining of gravel: 1. Narrow-channel runs,

J. Hydraul. Eng., 123(10), 874–884, doi:10.1061/(ASCE)0733-

9429(1997)123:10 (874).

Stillwater Sciences (2002), Dam Removal Express Assessment Models

(DREAM), Technical Report, 62p, October 2002, available at http://

www.stillwatersci.com/publications/DREAM.pdf,accessedinJune 2006.

Toro-Escobar, C. M., G. Parker, and C. Paola (1996), Transfer function for

the deposition of poorly sorted gravel in response to streambed aggrada-

tion, J. Hydraul. Res., 34(1), 35–54.

Wilcock, P. R. (1998), Two-fraction model of initial sediment motion in

gravel-bed rivers, Science, 280, 410–412.

Wilcock, P. R., and J. C. Crowe (2003), Surface-based transport model for

mixed-size sediment, J. Hydraul. Eng., 129(2), 120–128.

Wilcock, P. R., and S. T. Kenworthy (2002), A two-fraction model for the

transport of sand/gravel mixtures, Water Resour. Res., 38(10), 1194,

doi:10.1029/2001WR000648.

Wilcock, P. R., G. M. Kondolf, W. V. G. Mattews, and A. F. Barta (1996),

Specification of sediment maintenance flows for a large gravel-bed river,

Water Resour. Res., 32, 2911–2921.

Wu, F. C., and Y. J. Chou (2003), Simulation of gravel-sand bed response to

flushing flows using a two-fraction entrainment approach: Model devel-

opment and flume experiment, Water Resour. Res., 39(8), 1211,

doi:10.1029/2003WR002184.

Wu, W., and S. S. Y. Wang (2006), Formulas for sediment porosity and

settling velocity, J. Hydraul. Eng., doi:10.1061/(ASCE)0733-

9429(2006)132:8(858), 858-862.

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Y. Cui, Stillwater Sciences, 2855 Telegraph Avenue, Suite 400, Berkeley,

CA 94705, USA. (yantao@stillwatersci.com)

16 of 16

W10436

CUI: TUGS MODEL

W10436