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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

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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

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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).

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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

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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

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