ALLUVIAL FANS FORMED BY CHANNELIZED FLUVIAL
AND SHEET FLOW. I: THEORY
By Gary Parker/ Member, ASCE, Chris Paola,:1 Kelin X. Whipple/ and David Mohrig4
ABSTRACT: Alluvial fans and fan-deltas are of three basic types: those built up primarily by the action of
constantly avulsing river and stream channels, those constructed by sheet flows, and those resulting from. ~e
successive deposition of debris flows. The present analysis is directed toward the first two types. A mechamstic
formulation of flow and sediment transport through river channels is combined with a simple quantification of
the overall effect of frequent avulsion to derive relations describing the temporal and spatial evolution of mean
(i.e., averaged over many avulsions) bed slope and elevation in an axially symmetric fan. An example of a fan
formed predominantly by the deposition of sand is compared to a similar one formed predominantly by the
deposition of gravel. In each example the case of channelized flow is compared to the case of sheet flow. The
model is applied to the tailings basin of a mine in the companion paper.
INTRODUCTION
Alluvial fans are fan-shaped zones of sedimentation downstream of an upland sediment source. Fan shape tends to be
regular and conical, and can often be described as axially symmetric to a first approximation (Hooke and Rohrer 1979). At
least three mechanisms for their formation have been observed; avulsing channelized rivers, sheet flows, and debris
flows (Schumm 1977; Blair and McPherson 1994). In the case
of a sheet flow, channelized upland flow reaches the fan and
spreads out widely with no obvious channel, inundating much
of the fan surface and depositing sediment [e.g., Blair and
McPherson (1994)]. Debris flow fans are built up as an agglomeration of tongue-shaped deposits of individual debris
flows, each one of which is typically much smaller than the
fan itself [e.g., Johnson (1984); Suwa and Okuda (1983);
Whipple and Dunne (1992)]. Fluvial fans are built up by the
successive aggradation, and then avulsion of a river. The river
channel may be meandering, split-channel, or fully braided.
An example of a large fluvial fan is that of the Kosi River,
India, as it emanates from the Himalaya Mountains, shown in
Fig. 1 (Gole and Chitale 1966; Wells and Dorr 1987). Fluvial
fans may be completely terrestrial, or may have a distal portion
with standing water. Fans of the latter type are called fandeltas (Nemec and Steel 1988). Any given fan may be built
up by some combination of the previously mentioned mechanisms. It is fair to say, however, that fans dominated by debris
flows tend to have higher slopes than those dominated by fluvial processes [e.g., Harvey (1984); Wells and Harvey (1987);
Jackson et al. (1987)]. The present paper is devoted to fluvial
fans. The analysis also allows for the description of sheet flows
as a limiting case.
It is not accidental that fans often form in regions that are
undergoing subsidence in geologic time. Subsiding regions are
natural sinks for sediment; it is by this mechanism that sedimentary basins fill (Allen and Allen 1990). Subsidence, while
rarely exceeding rates of a few millimeters per year, neverthe'Prof.• St. Anthony Falls Lab.• Univ. of Minnesota. Minneapolis. MN
55414.
2 Assoc. Prof.• Dept. of GeoJ. and Geophys., Univ. of Minnesota. Minneapolis. MN 55455.
3Asst. Prof.. Dept. of Earth. Atmospheric. and Planetary Sci.. Massachusetts Inst. of TechnoJ.. Cambridge. MA 02139.
'Res. Sci.. Exxon Production Research Co.• Houston. TX 77252.
Note. Discussion open until March 1. 1999. Separate discussions
should be submitted for the individual papers in this symposium. To
extend the closing date one month. a written request must be filed with
the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 27. 1997. This
paper is part of the JourTUZl of Hydraulic Engineering. Vol. 124, No.
10. October. 1998. ©ASCE, ISSN 0733-9429/98/0010-0985-0995/$8.00
+ $.50 per page. Paper No. 15069.
less acts to limit the horizontal growth of fans, as shown by
the classic example from Death Valley (Hooke 1968, 1972).
As a fan builds outward in a subsiding basin of sufficient extent, it must eventually reach a point at which all of the sediment brought into its head is consumed in providing the deposit just necessary to balance subsidence, so that outward
progradation ceases. Under these circumstances the fan
reaches a state of equilibrium aggradation, at which the mean
aggradation rate due to sediment deposition just balances the
subsidence rate, and mean elevation on the fan remains constant in time. This balance was described in a mechanistic
sense by Paola (1988, 1989) and Paola et al. (1992).
Another mechanism for driving sediment deposition is rising base level, e.g., that of a lake or the ocean. This mechanism has particular relevance to fan-deltas. It is shown here
that the equilibrium aggradation associated with a base level
rising at a constant rate and vanishing subsidence is identical
to that associated with a constant rate of subsidence but invariant base level.
Whether actively prograding outward or in a state of equi87"10'
nKosi ft
su
Nepal
a:
~
'f,~I~'
Barahakahetra
Chalra
N
f
li.l..l..l..J
25Km
FIG. 1. View of Alluvial Fan of Kosi River, India, Showing Location of Main Thread of Flow at Various Times [Adapted from
Gole and Chitale (1966)]
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librium aggradation, fans with a range of sizes in the source
material usually display at least some degree of downstream
fining, Le., a tendency for characteristic grain size on the alluvial surface to become finer in the downslope direction [e.g.,
Blissenbach (1952); Shaw and Kellerhals (1982)]. Here, however, the analysis is restricted to uniform sediment for simplicity. The analysis of Cui et al. (1996) suggests a way in
which the gravel case can be generalized to grain mixtures.
It should be noted here that the achievement of equilibrium
aggradation by a fluvial fan by no means imparts any stability
to the river channel or channels on the fan. A constant mean
bed elevation at any point is achieved only by the shift or
avulsion of the river channel(s) over the entire fan, so as to
provide enough sediment to balance subsidence. Even in the
case of equilibrium aggradation, then, the channel(s) must
ceaselessly rework the entire fan surface whenever the flow is
sufficient to render it active.
This ceaseless reworking at geomorphic time scales is the
cause of numerous problems at engineering time scales
(French 1987, 1992; Dawdy 1979). A significant fraction of
the population of the Southwest United States lives on alluvial
fans. A river channel on such a fan may appear stable for
decades, and then completely relocate itself on the fan surface
after a single flood event, with adverse consequences for the
homes, roads, railways, and bridges so affected. A recent example of a bridge problem associated with an alluvial fan is
the SH6 bridge over the Waiho River, New Zealand (Thompson 1991).
In the case of natural fans, most of the activity that builds
a fan occurs during floods, Le., when the channel(s) is morphologically at its most active. The vagaries of flooding are
such that some can be associated with large sediment delivery
from the upland zone, and others with a smaller sediment delivery. As a result, parts of the fan may undergo degradation
from time to time, even though the fan is aggrading in the
long-term average. It has been postulated that fan buildup in
this case is dominated by a long-term quasi-cyclic "pumping"
process of sediment transfer. That is, during some periods sediment builds up in the proximal zone of the fan near its head,
with little sediment reaching the distal areas. During other periods this proximal deposit is partially downcut, with the sediment so yielded being delivered to the distal zone of the fan
and depositing there [e.g., Schumm (1977)].
It is of interest, however, to consider the simplest possible
case, i.e., one for which the discharge of water and sediment
at the fan head is constant in time, and for which the sediment
is uniform in size. This case is not only of value as a simplification of nature, but is also quite easily modeled in the laboratory [e.g., Hooke and Rohrer (1979); Schumm et al. (1987);
Whipple et al. (1995, 1996); Bryant et al. (1995)]. While the
simplicity of the upstream boundary conditions may rule out
a quasi-cyclic pumping of sediment, most other interesting features of fluvial fans are reproduced, including ceaselessly
avulsing channels and the evolution of a fan-shaped deposit
approaching axial symmetry. A mechanistic explanation of fan
morphology under these simple conditions would be a useful
step toward a predictive understanding of natural fan morphology.
Field examples of alluvial fans or fan-deltas that are not far
from the simple case described before are provided by the
tailings basins of mines. Efficient mine production requires
that the delivery of water and tailings to the tailings basin must
be quite constant from day to day. The grain size distribution
of the tailings, however, rarely approaches uniformity; while
the characteristic size is often in the range of sand or coarse
silt, there is often an admixture of mechanically produced
grains with sizes in the range of fine silt and clay. The fan
formed in the tailings basin can thus be expected to display at
least some degree of downstream fining. Indeed, the finest material may deposit only in standing water. If the water is to be
recycled with minimal sediment recycling, then a ponded zone
must be maintained along the distal edges of the fan.
Because mines operate on engineering, rather than geomorphic time scales, basin subsidence can be neglected in considering fan evolution. Since a properly constructed basin captures all sediment delivered to it without loss or recycling,
however, the fan in a tailings basin subject to a constant inflow
of water and sediment must eventually achieve a state close
to equilibrium aggradation. This is illustrated in Fig. 2 for the
West Area number 1 of the tailings basin of an iron mine in
northern Minnesota, called herein the "Rolling Stone" Mine
(Whipple et al. 1996). Because fan elevation is rising everywhere at a roughly uniform rate, the dikes that confine the
basin must be built up over time. The case thus corresponds
to fan formation under conditions of rising base level.
The present paper is one of a pair, with the companion paper
being that of Parker et al. (1998). The present paper is devoted
to the development of a theory of the evolution of an axially
symmetric fan under the constraints of constant water and sediment inflow during floods and constant sediment size. The
effect of varying hydrology is introduced in a simple way using an intermittency factor corresponding to the fraction of
time the channel(s) is in flood. The results are then applied to
the case of equilibrium aggradation. It is shown that the case
of equilibrium aggradation balancing subsidence such that
mean absolute elevation does not change in time is mathematically equivalent to the equilibrium aggradation occurring
in a confined basin such that bed elevation everywhere rises
at an average rate that is constant in time. The results of the
analysis are then used to explore the role of fluvial channelization in fan evolution. This is done by comparing the case
for which channelized flow is allowed to build the fan through
495
..... 485
~
i_
... 475
w
300
Distance (M)
FIG. 2. Bed Elevation Profiles for Various Years in West Area Number 1 of Tailings Basin of "Roiling Stone" Mine, illustrating Approximately Equilibrium Aggradation
986/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998
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avulsion against the case of pure sheet flow with no channelization. The assumption of constant water and sediment
discharge is relaxed in a simple way, with the use of an
intermittency, to allow for predictions that more closely approximate nature. As illustrated next, the theory predicts that
channelization acts to significantly reduce fan slope compared
to what would prevail under sheet flow.
In the companion paper, Parker et al. (1998), the theory is
applied to the engineering problem of the fan in the tailings
basin of the "Rolling Stone" Mine. The assumption of uniform sediment is modified in a simple way to account for the
grain size distribution. The analysis leads to an engineering
tool for testing schemes for prolonging the lifetime of the tailings basin.
The present paper is also closely related to two other papers,
those of Whipple et al. (1998) and Paola et al. (1998). The
former paper is devoted to a study of experimental fans formed
in the laboratory; the experimental results are compared
against the theory presented here. In the latter paper the role
of shear stress variation within the channel(s) is used to develop corrections for sediment transport and resistance relations for channelized flow on fans. These corrections reflect
the diversity of flow conditions associated with meandering or
braided stream morphology.
EXNER FORMULATION FOR FAN EVOLUTION
The geometry of Fig. 3 is considered. In plan view the fan
is assumed to have a conical shape with angle 6 and radius L.
The fan width Bf defines an arc given as a function of the
radial coordinate r as
(1)
The basin itself may be undergoing subsidence at a vertical
speed V s ' Here, the subsidence rate is taken to be constant and
uniform [e.g., Whipple and Trayler (1996)], although the theory is easily generalized to spatially and temporally varying
subsidence rate. The volume discharge of water Qw, the feed
rate of sediment Qso at the head of the fan, and the grain size
D of the feed sediment are assumed to be constant for fraction
I of time, where I is an intermittency, and vanishing for fraction (1 - 1) of time (Paola et al. 1992). Loss to ground water
is neglected, so that the total water discharge Qw crossing the
arc normal to r is assumed to be constant everywhere. The
total sediment discharge Qs crossing the arc normal to r is not
constant, however, because the fan is built up by deposition
associated with a downstream decline in sediment discharge.
The porosity Ap of the deposited sediment is assumed to be
constant. It is assumed that no sediment escapes the basin, so
that the following boundary conditions apply:
(2a,b)
It is assumed that at any given time during which there is
flow, one or more active river channels course the fan. The
present description does not distinguish between single-channel and braided configurations; Bac simply denotes the total
width of all the active channels measured along an arc normal
to r. Thus
(3)
where qs = mean volume sediment transport per unit width in
the channels. As these channels deposit sediment, they aggrade
and thus shift or avulse to regions of locally low elevation on
the fan, so building up the fan across its entire width Bf . The
present formulation is averaged over a time scale that is long
compared to the time necessary to rework the entire fan surface at least once. Under this condition and the condition of
axial symmetry, the mean bed elevation 1') at a point can be
taken to be a function in space of r alone, with no angular
variation.
Under the foregoing constraints, during the fraction I of
time for which flow occurs on the fan the Exner equation of
sediment continuity takes the following form:
aT]
( 1 - A)B ( p
:t at
+ v s)
aQs
=-ar
(4a)
Subsidence continues even during the fraction (1 - I) of time
for which there is no flow, during which time the Exner equation takes the form
(1 - Ap)Bf
e~ + vs) = 0
(4b)
The two prior equations can be averaged over time by multiplying (4a) by I, (4b) by (1 - I), and adding. After some
reduction the following time-averaged result is obtained:
aT]
(1 - A)B ( p
:t at,
+ -vs)
= -aQs
I
ar"
t
= It
(4c,d)
In (4c,d) 1') now = hydrologically averaged bed elevation; and
t. = effective time. The formulation has been adapted from the
original formulation of Paola et al. (1992). Two subcases are
of particular interest here. The first, which is of general geological significance, is that of a natural fan for which a perfect
balance between aggradation and subsidence has been
achieved, so that fan shape averaged over many avulsions no
longer evolves in time. In this case bed elevation 1') becomes
constant in time and (4c) reduces with (1) to
d
dr (q sB ac )
-
=-dQs
dr = -(1
vs
- A)
-I 9r
p
(5)
Imposing (2a,b) and solving, it is found that
g;. [I - (£)']. T ~ ~:,)9L'
=
=
(I
(6a,b)
It is seen that the sediment transport rate Qs must decrease
FIG. 3.
Definition Sketch for Conical Fan
parabolically down the fan in this case. Since the subsidence
rate V s can be considered as the first order to be imposed by
tectonic effects [e.g., Allen and Allen (1990)], (6b) indicates
that basin radius L is a dependent variable.
Interpreted another way, for an imposed subsidence rate v"
(6b) implies that a fan can prograde no farther from its head
than distance L. Once this length is reached, all the sediment
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delivered to the head is consumed in balancing subsidence,
leaving nothing remaining to drive further progradation.
The second subcase has engineering as well as geomorphic
significance, as shown in the companion paper, Parker et al.
(1998). Here the subsidence rate is assumed to be zero, or at
least negligible over the engineering time scale. The basin is
assumed to be confined by dikes along its sides and at its
downstream end; water is allowed to escape (perhaps to be
recycled) but no sediment escapes. During periods of flow the
dikes are assumed to be built up in time at a constant vertical
speed Vd so as just to contain the sediment. During periods
with no flow the dikes are not built up. The geometry thus
corresponds to a tailings basin of a mine. The state of equilibrium aggradation is achieved everywhere when the bed builds
up at the same speed Vd as that of dike-building, so that
channel shear velocity, which is, by definition, related to the
average boundary shear stress Tb as follows:
(11)
where p = water density. It is assumed that
!:!- = a, (!!.)P
u*
(12a)
D
where a, = dimensionless coefficient; and p = dimensionless
exponent. The choice p = 1/6 yields the original Manning relation; the choice p = 0 yields a Chezy resistance relation. It
must be recognized that in any given case (12) would describe
a mixture of skin friction and form drag in some unspecified
ratio. It is useful to express the coefficient a, as follows:
(7)
where t. is given by (4d); 'TILo = elevation at the downstream
end of the basin at t = 0; and basin elevation 'TI, relative to the
elevation at the downstream end at any time is taken to be a
function of r alone. Setting V s equal to 0 in (4a) and reducing
with (1) and (7), it is quickly found that
d
dr (q s B oc )
-
=-dQs
dr = -(I
Vd
- A) P
I 6r
(8)
and thus with the imposition of (2a,b)
(9a,b)
Note that (9a,b) are essentially identical to (6a,b); the only
difference being that the subsidence rate v, is replaced with
the rate of dike raising Vd' There is, however, a conceptual
difference. In the case of natural equilibrium aggradation balancing subsidence, v, is an independent variable and basin
radius L is a dependent one. The situation is reversed in the
case of a tailings basin, where the height of dike raising Vd
must be chosen just so that no sediment overspills the dike.
This condition corresponds to an optimal engineering solution,
as it serves no purpose to build the dikes higher than necessary.
The analysis immediately prior is directly applicable to the
case of a natural fan-delta that is responding to base level
rising at speed Vd rather than subsidence.
INTERNAL RELATIONS
Further progress requires the specification of internal relations describing the flow and sediment transport in the channels. In particular, three relations are required. The first describes flow resistance in the channels, the second describes
sediment transport, and the third specifies the total width of
the active channels Bac • Here general relations of simple power
fonn are adopted, in that they can be used to obtain analytical
solutions for equilibrium aggradation. The framework of the
theory is sufficiently flexible, however, to encompass quite
general internal relations.
While in reality the flow on the fan may be single-channel
or braided at any given time, all the channels are here lumped
into a single "effective" channel with average depth H, flow
velocity U, and sediment transport per unit width qso (Fig. 3),
such that
(10)
The statistical implications of this procedure are discussed
next. The relation for flow resistance is assumed to be of the
generalized Manning-Strickler fonn. Let u* denote average
(12b)
where a,o = resistance coefficient that would prevail in the
case of flow in a straight flume-like channel; and adjustment
coefficient a,a = order-one factor providing bulk accounting
for the variability associated with an actual meandering or
braided channel [Paola (1996); Paola et al. (1998); see text to
follow].
A generalized sediment transport law of the form of MeyerPeter and Muller (1948) is assumed here as
q,
= a ('1'* - T*)n
VRgDD'
C
Here g = gravitational acceleration; as
ficient; n = dimensionless exponent
R
=-p,P -
1,
'1'*
(13)
= dimensionless coef-
Tb
=pRgD
(l4a,b)
and T~ = critical Shields stress associated with the threshold
of sediment motion. The exponent n takes a value of 1.5 in
the original relation of Meyer-Peter and Muller (1948), which
was obtained for the case of relatively uniform gravel. In the
case of the transport of sand at conditions well above the
threshold of motion, however, the formulation of Engelund
and Hansen (1972) has an exponent of 2.5. The van Rijn
(1984a,b) relation for sand can also be locally fitted to the
form of (13) for the same transport conditions, yielding values
of n that are again substantially higher than 1.5 (Whipple et
al. 1998).
As was done for the resistance relation, the coefficient as is
expressed as
(14c)
where a,o = coefficient for the case of a straight, flume-like
channel; and a,a = order-one coefficient quantifying the bulk
effect of variations in meandering or braided channels (Paola
1996; Paola et al. 1998). In general, a sa can be expected to be
greater than unity. This is because in any river such parameters
as the bed shear stress vary locally about their mean values,
being high in some places and low in others. The nonlinearity
in any typical sediment transport relation is such that zones of
high shear stress, for example, should contribute disproportionately to the sediment transport. The effect in bulk is an
increased rate of sediment transport relative to that which
would be predicted based on the locally constant shear stress
in a flume-like channel.
It is seen from both (12) and (13) that knowledge of an
average boundary shear stress Tb is required. Here it is assumed
that nowhere does the flow deviate too strongly from equilibrium, or normal conditions, so that local equilibrium momentum balance reduces to
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Tb
= pgHS
(15)
where S = channel slope. Since Tb and H are bulk values characterizing the flow in the channels and local equilibrium is
assumed, S may also be identified with local average fan slope
(averaged over many avulsions), Le.
S=
_a'T)
ar
(16)
In the case of a relatively steep fan with braided channels, the
Froude number of the flow may be high enough to render this
crude assumption of local equilibrium reasonable as a firstorder approximation. It is likely to be less accurate when the
slope of the fan is low and a single, meandering channel prevails. In the latter case, a more correct formulation may thus
require a description based on the assumption of gradually
varied flow.
The final relation is the one governing the width Bac across
which sediment is transported by the flow. In the extreme case
of pure sheet flow inundating the entire fan, Bac is simply
given by fan width, i.e.
(17a)
A more common case can be described as fractional inundation
by sheet flow. Here this is treated in terms of a spreading angle
that is a set fraction of fan angle e, so that B ac is given by
(17b)
where X = constant between 0 and 1. While such a flow covers
only a portion of the fan at any given time, it is assumed to
periodically shift so as to lead to deposition over the entire
fan in the long term. The experimental study of Whipple et
al. (1998) addresses the issue of fractional sheet flow.
Another case of interest here is that of a channelized, or
fluvial fan. In the case of a meandering channel Bac is simply
the mean width of the channel over which sediment is actively
transported; in the case of a braided channel it is the mean of
the sum of the active widths. Here the assumption introduced
to close the problem does not even directly involve Bac • It is,
instead, assumed that channels evolve on the fan in such a
way that the "effective" channel maintains a constant Shields
stress
associated with active, mobile-bed conditions. From
(l4b) and (IS), then, the relation for channel form can be
expressed as
T:
H
- S=
D
CXb
'
CXb
= RTa*
(l8a,b)
where CXb is taken to be a constant. That this does, indeed, lead
to a closure for B ac is demonstrated next.
The form of (18) is easily justified as an approximation in
the case of active self-formed channels with gravel bed and
banks. Parker (l978a) provided a theoretical justification for a
value of T: equal to about 1.2 times the threshold value T~;
field data indicate a value closer to 1.4. Paola et al. (1992)
have previously used such a formulation in a model of basin
filling by braided gravel-bed channels that serves as a prototype for this one. In the case of sand-bed streams the justification must be at least partially empirical. The general form
of the relation in terms of (HID) and S is suggested by the
relation obtained by Parker (1978b) for sand bed streams
H
m
D S = cx
cp
Somewhat more general significance can be attached to (18)
for the case of sand-bed streams by considering the compendium of data on rivers, by Church and Rood (1983). Only
sand-bed streams with a characteristic grain size D less than
0.5 mm are considered here. 1\\'0 data subsets were extracted
from the compendium. The first pertains essentially to single
channel meandering streams (sinuous channel, meandering, or
with minor secondary channels). The second pertains to multiple channel streams (occasional split channels to braided
channels). Flow conditions were chosen to be formative, Le.,
bankfull; where this was unavailable, the two-year flood was
chosen. Paved or degrading channels were excluded, as were
channels with nonalluvial banks. Seven single channel streams
and 11 multiple channel streams were selected in this way. In
Fig. 4 the Shield stress
at formative discharge is plotted
against slope S. Also included in the plot are data corresponding to averages for three large subsets of measurements in
active sand-bed braid channels in the tailings basin of the
"Rolling Stone" Mine, which are explained in more detail in
the companion paper.
While the data display significant scatter, the values of
show no significant trend in slope S, lending credence to the
The
crude approximation of a roughly constant value of
overall average for
for all 20 points is 1.83. If two anomalously low points (multiple-channel streams) and one anomalously high point (single-channel stream) are excluded, all the
:s; 3.05, with an average
data fit within the band 1.33 :s;
value of 1.79. More specifically, average values for each subset
(after excluding the three anomalous points) yield the following estimates for T:: single-channel streams, 1.72; multiple
channel streams, 1.84; "Rolling Stone" Mine tailings basin,
1.81. Within the scatter of the data, an overall crude estimate
of
of 1.8 would appear to be appropriate for alluvial sandbed streams of all planforms but with a characteristic size D
below 0.5 mm.
It is not implied here that all, or even most, of the reaches
of the streams in question are on alluvial fans. The purpose of
Fig. 4 is rather to demonstrate that (18) is a crude but not
unreasonable assumption for sand-bed streams in general. A
comparison of the consequences of the closure assumptions
(17) and (18) for the active width of sediment transport allows
for a comparison of the morphology of channelized fluvial fans
with that of fans produced by pure or partial sheet flow.
The choice of internal relations may be dependent upon,
among other things, scale. For example, the fan in the tailings
basin of the "Rolling Stone" Mine is channelized into a
braided configuration. The vastly smaller scale of the experimental fans discussed in Whipple et al. (1998) leads to a configuration better described by fractional sheet flow. The sediment transport and resistance relations can be chosen to
describe the case of interest. Perhaps the least reliable of the
T:
T:
T:.
T:
T:
T:
10.00
100
..
. .-
• multi
• single
• "RS"
,;
(19)
where m = 0.6; and cx cp = dimensionless function of a Reynolds
number based on particle diameter D. An empirical justification for (18) for a specific case is presented in the companion
paper.
0.10
1.00E-03
1.00E-05
1.00E-02
S
FIG. 4. Values of Shields Stress
tions for Various Rivers
T: under Formative Condi-
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internal relations is the one describing the Shields stress of
sand-bed streams at fonnative flow.
SOLUTION FOR FAN UNDERGOING EQUILIBRIUM
AGGRADATION
Eqs. (10)-(15) can be reduced to the following general expressions for Shields stress T* and sediment transport rate Q.
on the fan:
2/(3+2P)
T* =
R-Ia;2/(3+2p)S(2+2P)/(3+2pl
(
Qw
)
ViDDB ac
[(..!..
Q.o(l - ,.2) )lIn
a.yRgDDx6L,.
,
+
T*]}(3+2p)/(2+2pl
c
Qw
)-l/(l+P)
Vii5Dx6Lf
(27)
,.= !..L
[R-Ia-2/(3+2plS(2+2p)/(3+2P) (
.,
Q.
1
S=R
{
)
-
Qw
)2/(3+2pl
ViDDBac
_
T*]"
C
(21)
P
] }(3+2p)/(2+2 )
11"
[ ( a. yRgDDB ac
-II(1+Pl
+T~
Qw
'a1/(l+p)
( Vii5DB )
ac
,
(22a)
In the case of fractional sheet flow, (17 b) can be substituted
directly to eliminate B ac from (20)-(22), yielding
-
= {R
where
These fonns are independent of the assumption for the effective width of the active channels B ac • They specify local average Shields stress T* and sediment transport Q. as functions
of water discharge Qw and local average slope S. Eq. (21) may
be inverted to give S as a function of Qw and Q.
S-
S
'al/(l+p) (
(20)
Q.
=a
tion and subsidence and that pertaining to a balance between
deposition and dike raising (or base level rise), results in the
same mathematical fonn for the downstream variation of slope
S on the fan. For the case of partial sheet flow, substitution of
(6a) and (17b) into (23) yields
1 / " ] }(3+2P)/(2+2pl
R..!..
{
[ (
+ T~
Q.
a.YRgDDx6r )
-II(1+P)
'a1/(I+p)
(
r
Qw
Vii5Dx6r )
(22b)
The case of pure sheet flow is realized by setting X equal to
unity.
In the case of channelized flow, (21) can be reduced with
(18) to yield
Bac
D
= a;(3+2pll2a ;IS(I+Pl
(vfi
2)
(23)
gDD
or reducing further with (22a)
Bac _ -I
D - a.
Q.
(
yRgDD2
(a *)-"
)
b
R
_
Tc
(24)
Q.
~,;;-;::
v RgDD
_
2 -
-(3+2p)/2
a.ab
-I ( a b _
a,
R
*)"
Tc
S(l+P) (
Qw )
r-;:: 2
vgDD
•
(25)
S ={R-1I2a;la~+2P)/2a,
(ex; _T~ )-n [ Q.o(~~ ,.2)]}
= [ R-1I2a;la~+2p)/2a, (~ _ T~ ) -" (~:) ] I/(I+p)
Relation (22b) (for an unchannelized fan) or relation (25)
(for a channelized fan) can be substituted into the Exner equation [(4a)] and solved numerically to describe the time growth
of a fan under the boundary conditions (2a,b). Here, however,
interest is focused on the analytical solution that can be obtained for the state of equilibrium aggradation outlined previously.
Either of the two cases of equilibrium aggradation considered earlier, i.e., that pertaining to a balance between deposi-
(29)
OBSERVATIONS ON ROLE OF CHANNELIZATION
Channelization is thought to play a fundamental role in the
development of a fluvial fan morphology that is distinct from
that created by sheet flows [e.g., Schumm et al. (1987)]. The
nonlinearity of sediment transport relations is such that for the
same discharge more sediment can be carried by a relatively
narrow, deep channel than a relatively wide, shallow one. Concomitantly, if the sediment discharge is held constant, the
channelized flow, being the more efficient transporter of sediment, should maintain a lower slope. This lower slope can be
imposed on the entire fan by the mechanism of avulsion. The
implication is that channelization acts to lower the slope at
each point on the fan below the value that would prevail for
unchannelized sheet flow.
A rather interesting limiting case serves as a foil for a discussion of the effect of channelization on fan morphology. Let
p = 0 (Chezy resistance relation), n = 3/2 (Meyer-Peter Muller
sediment transport relation), and let the Shields stress T* satisfy the condition
»
T~
(30)
Here T* is given by (22) and (17 b) for the case of partial sheet
flow and T* = T: according to (18) for channelized flow. Under
these conditions it is easily shown that (22) and (26) reduce
to the same relation for fan slope
= R a,
Q.
a. Qw
(26)
l/(l+p)
The elevation profile 'Tl(r) is found by integrating (27) or
(29) subject to the boundary condition of a constant known
elevation at the upstream end of the fan in the case of deposition balancing subsidence, and a known elevation rising linearly with speed Vd at the downstream end of the fan in the
case of a tailings basin (or rising base level).
S
or inverting for S as a function of Qw and Q.
S
For the case of channelized flow, substitution of (60) into (26)
yields
T*
Using (13), (18), and (24), the relation for sediment transport
Q. as a function of Qw and S becomes
(28)
(31)
That is, slope obeys a simple linear dependence on the ratio
of the sediment discharge to the water discharge. Note that the
coefficient for width closure ab drops out of the fonnulation
for channelized slope in this case. Were the stated assumptions
to hold true in general, channelization would play no role in
reducing fan slope below the value that would prevail for sheet
flow.
Exploration of the effect of variation of the parameters in
(22) and (26) allows for a detennination of the role of channelization. The exponent p in the resistance relation is generally found to be rather small compared to unity (the standard
990 I JOURNAL OF HYDRAULIC ENGINEERING I OCTOBER 1998
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Manning value being 116), so a relaxation of the assumption
p = 0 plays little role in regard to the effect of channelization.
If the exponent n in the sediment transport relation is kept
equal to 3/2 but the Shields stress is no longer constrained to
satisfy (30), it is found that the slope of a channelized fan
drops below the corresponding unchannelized case. If, on the
other hand, (30) is retained but n is allowed to increase beyond
3/2, it is again found that the slope of the channelized fan
drops below the corresponding unchannelized case.
Each scenario has its expression in the real world. The
Meyer-Peter and Muller (1948) sediment transport relation is,
perhaps, the most reasonable of the simpler sediment transport
equations for gravel transport. As noted before, however, selfformed gravel-bed channels with gravel banks tend to maintain
Shields stresses that are not far above the critical value for
motion. As illustrated by the following example problem, the
result is channelized fan slopes that are significantly less than
their unchannelized counterparts.
Self-formed sand-bed streams typically maintain Shields
stresses at formative (i.e., bankfull) flows that are greatly in
excess of the threshold value for transport. A simple fit of the
more reliable relations for the transport of sand into the form
of (13), however, yields values of n in excess of 1.5 in the
limit of high Shields stress. A simple illustration of this can
be given with the total sand transport relation of Engelund and
Hansen (1972). It takes the form
(32)
where Cf = friction coefficient given by the relation
Cf 1/2
_
-
U
(33)
u*
The value of n is seen from (32) to be 2.5; comparing with
can be effectively set equal to zero.
(13) it is seen that
(This does not imply that there is no threshold of motion in
the Engelund-Hansen formulation; rather, it implies that it does
not appear explicitly in the load equation, itself.) Upper-regime
flow is taken as an example here, as it can be expected to be
rather common on the relatively steep slopes of alluvial fans.
In this case the form drag associated with dunes can be neglected, and C;1I2 can be expressed as a logarithmic, and thus
slowly varying, function of HID. Taking the rather typical
value for C;1I2 of about 15, (32) yields a value of a so of 11.25.
Similar approximate expressions for nand a so can be obtained
by means of a local fit of the sand transport relations of van
Rijn (1984a,b) to (13) (Whipple et a1. 1998).
The foregoing observations are used as the basis for a more
specific calculation. A subsiding basin with a radius of 10,000
m and an angle of 120° containing a fan undergoing equilibrium aggradation in balance with subsidence is considered.
The specific gravity of the sediment delivered to the head of
the fan is assumed to be 2.65, yielding a value of R of 1.65.
The porosity of the deposited sediment Ap is assumed to be
0.4. These values are given in Table 1.
Two cases summarized in Table 1 are considered-one for
a gravel fan and the other for a sand fan. In both cases the
fan produced by channelized flow is compared to the one resulting from pure channelized sheet flow covering half the fan
surface during any given flood (X = 0.5). In both cases a Chezy
resistance relation, according to which p = 0 and aro =
C;1I2, is postulated. In the case of the gravel fan characteristic
grain size D is taken to be 20 mm and C;lt2 is set equal to
10. The Meyer-Peter Muller sediment transport relation is assumed, so that n = 1.5 and a so = 8. The critical Shields stress
is set equal to 0.03 and the Shields stress of the active
is set to be 1.4 times
or 0.042, yielding a value
channels
T:
T:
T:
T:,
TABLE 1. Assumed Parameters for Calculation of Fan Characteristics
Parameter
(1 )
Gravel fan
(2)
Sand fan
(3)
Notes
(4)
L (m)
6
10,000
2.094
1.65
0.4
20
0
10
1
1.5
8
3.0
0.030
0.042
0.0693
200
0.1
0.02
1.00
1.67 x 10'
10,000
2.094
1.65
0.4
0.30
0
15
1
2.5
11.25
1.5
0
1.8
2.97
20
0.04
0.05
1.00
1.67 X 10'
Basin radius
120·
Specific gravity
Porosity
R
lI.p
D (mm)
p
IX",
IX..
n
IX.o
IX••
,.*.
"~
IX.
Qw (m'/s)
Q.o (m'/s)
I
v. (mrn/yr)
y (ttyr)
=2.65
Chezy law
2
= Ci"
-
-
=,.: R
-
-
Intermittency
Subsidence rate
Sediment yield
of ab of 0.0693. Insofar as gravel rivers tend to be rather
flashy, especially on fans, flow is assumed to occur only 2%
of the time, or 7.3 days per year, yielding an intermittency
factor I of 0.02. This value may be reasonable for humid and
subhumid fans, but high for desert fans. During the time of
flow the water discharge Q", is assumed to be 200 m 3/s, and
the sediment feed rate at the head of the fan is assumed to be
0.1 m3/s. These numbers yield a subsidence rate V s of 1.00
mm/yr in accordance with (9b) and a sediment yield to the
basin Y of 1.67 X 105 t/yr.
In the case of the sand fan, a value of grain size D of 0.30
are set equal to IS,
mm is selected; aro = C;It2, n, a so , and
2.5, 11.25, and 0, respectively, in accordance with the prior
discussion. In accordance with the discussion associated with
Fig. 4
is set equal to 1.8, yielding a value of ab of 2.97 for
sediment with a natural specific gravity of 2.65. This value of
active channel Shields stress is well in excess of the threshold
for the motion of sand, which should not exceed 0.06.
value
Insofar as sand-bed streams typically have lower slopes than
gravel-bed streams, they tend to have a longer time of concentration and are thus less flashy, at least in humid and subhumid regions. The intermittency I is thus set equal to 0.05,
corresponding to 18.3 days of active flow per year. The value
would be much lower for a sandy desert fan. The sediment feed
rate Qso during the period of flow is set equal to 0.04 m 3/s, so
as to give the same subsidence rate V s and sediment yield Y
as the gravel fan. Water discharge Q", is set equal to 20 m 3/s
so that the concentration of sediment entering at the head of
the sand fan is four times that of the gravel fan.
In both cases the morphologic adjustment coefficient a'a in
the resistance relation defined by (12a) is set equal to unity.
This reflects the result of Paola (1996) and Paola et al. (1998)
showing that the correction in the resistance relation is not
large. More problematic is the morphologic adjustment coefficient asa in the load relation defined by (14c). The appropriate value for a flume is unity. The value of asa for a meandering channel can be expected to be in excess of unity. A
comparison of sediment transport in flumes and field rivers
(most of which were meandering) led Brownlie (1981) to propose a multiplicative adjustment coefficient corresponding to
asa of 1.268 in his own sediment transport relation. This value
may be an underestimate in that the measured sediment transport rates typically pertain to flow conditions in relatively
straight reaches of meandering streams, rather than reach-averaged flow conditions. In the case of a braided channel asa
can be yet higher; Paola (1996) and Paola et a1. (1998) indicate
T:
T:
T:
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that values in the range of 2-4 can be realized. The implication is that local variability in flow and morphology tends to
increase the effectiveness of sediment transport.
Here only loose estimates need be employed. In the case of
a gravel fan this coefficient is set equal to 3.0, a value that
would appear to be in the right range for braided gravel-bed
channels based on the work of Paola (1996). In the case of
the sand fan the value is lowered to 1.5, in anticipation of a
morphology closer to meandering, and in light of the work of
Brownlie (1981) mentioned earlier. The choice of a lower
value for exsa for the sand fan than for the gravel fan based on
an expected difference in morphology is an assumption that
can be tested a posteriori based on the predicted values for the
aspect ratio of the active channel(s) BaclH.
The results of the calculations for the gravel fan are shown
in Figs. 5(a)-(c), in which the spatial distributions of fan slope
S, width of the active channels BaC' and depth H for the case
of unchannelized sheet flow are compared with the channelized distributions. (Recall that Bac = XBf for the case of sheet
flow; in this example X takes a value of 0.5.) In Fig. 5(a),
unchannelized slopes are on the order of 0.03, or an order of
magnitude larger than channelized slopes. The channelized
slopes are seen to be fairly constant near the upstream end of
the fan and dropping toward the downstream end. This implies
an elevation profile that is straight near the upstream end and
concave near the downstream end. In Fig. 5(b), unchannelized
active widths are equal to half the fan width Bit and take values
approaching 10 km at the downstream end of the fan. The
channelized active widths are on the order of 200 m, and decline in the downstream direction as a result of deposition. In
Fig. 5(c), unchannelized depths quickly drop off to the order
of centimeters, whereas channelized depth is on the order of
1 m and increasing in the downstream direction. The effects
of channelization on the fan, as manifested through values of
T: that are only modestly above the critical value T:', are dramatic.
The results of the calculation for the sand fan are equally
dramatic, as shown in Figs. 6(a)-(c). In Fig. 6(a), unchannelized slopes are on the order of 0.01, whereas channelized
slopes are around 0.001 and declining in the downstream di0.1
0.01
0.01
•• Sheet
-Channel
0.001
II)
•• Sheet
-Channel
0.001
0.0001
00001
L.-_ _'---_---l_ _-----'_ _~_ ___'_J
o
0.2
0.6
0.4
0.00001
0.8
0
r/L
0.2
0.4
0.6
0.8
r/L
10000
1000
1000
- • Sheet
l-ch<lnnell
u
m
I
u
m
•• Sheet
i-Channel
100
100 •
10
0.2
0.6
0.4
0.8
r/L
10
§:
:I:
- • Sheet
-Channel
0.1
0.01
0.001
o
r/L
FIG. 5. Profiles of Channelized and Unchannelized: (a) Slope
Sfor Gravel Fan; (b) Active Channel Width S •• for Gravel Fan (In
Unchannelized Case, Width Is Half Fan Width S,); (c) Depth Hfor
Gravel Fan
0.2
0.4
0.6
0.8
r/L
FIG. 6. Profiles of Channelized and Unchannelized: (a) Slope
S for Sand Fan; (b) Active Channel Width S •• for Sand Fan (In
Unchannelized Case, Width Is Half Fan Width S,); (c) Depth H for
Sand Fan
992/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998
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rection. As in the case of the gravel fan, the elevation profile
of the channelized sand fan can be expected to be straight near
the upstream end and concave near the downstream end. In
Fig. 6(b), channelized active width Bae is seen to be around
20 m and declining in the downstream direction as sediment
is deposited. The predicted values are much less than the width
of the sheet flow. In Fig. 6(c), channelized depths are seen to
be on the order of 1 m and increasing in the downstream
direction, whereas unchannelized depths quickly drop to the
order of centimeters. Here the difference between the channelized and unchannelized cases is mediated by the degree to
which the exponent n in the sediment transport equation exceeds the value of 1.5.
The depth profiles for both the gravel and sand fans indicate
that a sheet flow at the specified discharge covering half the
fan surface would become so shallow as to be unsustainable
beyond 30% or 40% of the fan length. Calculations of this
type can provide useful bounds on the spatial extent and discharge of sheet flows inferred from outcrops.
The channels of the sand fan must exhibit a strikingly different morphology from those of the gravel fan. This can be
inferred from Fig. 7, where the ratio BaJH is plotted versus
rlL for the channelized gravel and sand cases. The value of
this ratio is around 300 on the upper part of the gravel fan,
but only around 40 for the sand fan. Incipient braiding is
known to be associated with a high width-depth ratio [e.g.,
Blondeaux and Seminara (1985)]. Appropriate estimates of the
value of channel width to depth for incipient braiding are in
the range of 100-180, with the lower value corresponding to
a higher slope [e.g., Hokkaido (1987)]. Evidently the gravel
fan must be strongly braided over the upstream half of the fan.
The width-depth ratio suggests that it should grade to wandering and then single-thread toward the distal end. The sand
fan can be expected to be single-channel meandering or sinuous throughout most of the fan.
It should be noted that the assumption for channelization
employed here breaks down at the downstream end of the fan.
According to (26), slope S must vanish at the downstream end
of the fan; (18) then implies that channel depth H must tend
to infinity there. Evidently the channel closure hypothesis fails
on extremely low slopes, a result that is, perhaps, not unexpected. In the case of a fan-delta, the channels should give
way to sheet flow over deltaic lobes at the extreme distal end.
The predicted elevation profiles for the channelized gravel
and sand fans are shown in Fig. 8(a). As suggested earlier, the
upstream half of both is approximately linear and the downstream half of both is upward concave. The elevation profiles
of the unchannelized fans are shown in Fig. 8(b). The profile
of the gravel fan is upward convex everywhere, and the profile
for the sand fan is, likewise, upward convex everywhere except near the distal end, where a slight concavity is seen. Nei-
16 , - - - - - - - , - - - - - - - - - - - - - - - - - - .
14
12
1- - Sand
-Gravel
4
ther of these profiles is seen in the field [e.g., Bull (1977)], an
observation that may be useful in interpreting outcrops.
SUMMARY
The present analysis is, of necessity, quite simplified. Since
the channels are not explicitly modeled, the predictions for fan
morphology represent only averages. The model would require
considerable generalization before it could describe the cutand-fill sequences associated with strongly varied water and
sediment supply (Schumm 1977).
The generalization of the model to include downstream fining would be more straightforward. For the case of gravel, for
example, the simple bulk formulation of the type of MeyerPeter and Muller (1948) used here could be replaced by a
relation that explicitly includes grain size variation [e.g.,
Parker (1990, 1991a,b)]. While the analysis here focuses on
equilibrium aggradation, the treatment could be used to describe .the time-varying transient state toward equilibrium aggradation, as well. In the case of a fan-delta, this transient state
is characterized by a distinct prograding front, a feature easily
added to the present formulation.
Two objections can be raised against the use of a constant
Shields stress
for the active channels. While the assumption
of a constant value of
that is not far above the threshold
~al~e fo~ sediment motion has both theoretical and empirical
JustificatIOn for gravel channels, within any agglomeration of
individual active braid anabranches
can be expected to vary
probabilistically. Paola (1996) and Paola et al. (1998) have
included this feature, which results in an adjustment coefficient
in the resistance relation elTa that is typically close to unity and
an adjustment coefficient in the load relation elsa that can be
as large as three times unity or more, depending upon morphology. The present theory has been formulated so as to accommodate these modifications.
The second objection pertains to sand-bed channels. In this
case the assumption of a constant value of
well above the
T:
100
- - Sand
-Gravel
10
1\-------~--_+_--_I_----'-----!.j
o
0.2
0.4
0.6
08
r/L
FIG. 7. Plot of BaclH versus rlL for Channelized Gravel and
Sand Fans
T:
T:
T:
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threshold of motion represents pure empiricism. The data necessary to validate it for the specific case of the fan of the
"Rolling Stone" Mine are presented in the companion paper,
Parker et al. (1998). A justification (or disproof) of the relation
for sand-bed streams should be a goal of future research; perhaps Parker (1978a) provides a starting point.
Finally, in applying the work to desert fans, the neglect of
the loss of water from the channel(s) due to infiltration may
not always be justified. While it is relatively easy to modify
the theory to incorporate any specified pattern of infiltration,
the process is quite difficult to predict a priori. Suffice it to
note here that the loss of water is likely to suppress the tendency for profile concavity down the fan.
CONCLUSIONS
A mechanistic model is formulated for the evolution of axially symmetric alluvial fans created by sheet flows and rivers.
The analysis pertains to constant water discharge Q and volume sediment inflow rate Q.o, and uniform grain size D. The
effect of varied discharge and sediment supply is introduced
in a simple way by means of an assumed intermittency of flow.
In the case of sheet flow, the flow is assumed to inundate some
set fraction of the fan width with no channelization. Pure sheet
flow is realized when this fraction takes the value of unity;
otherwise fractional sheet flow is realized. The zones of deposition of successive fractional sheet flows are assumed to shift
so as to result in deposition across the entire fan. In the case
of a channelized fluvial fan, the active channel(s) is assumed
to maintain a Shields stress near a constant value of ,.:. While
at any instant the flow is assumed to be channelized, deposited
sediment is spread across the width of the fan in order to
describe the consequences of channel shift or avulsion. While
the analysis can be used to describe a fan evolving under general conditions, here attention is focused on the case of equilibrium aggradation in balance with tectonic subsidence of the
basin or rising base level.
The analysis indicates that channelization on fluvially dominated alluvial fans has the effect of dramatically reducing fan
slope as compared to pure unchannelized sheet flows. While
both the gravel and sand fans display this same feature, it is
mediated by somewhat different mechanisms. In the case of
gravel, the lowered slope is driven by a Shields stress in the
active channel(s) that is allowed to exceed the threshold value
for the movement of sediment only by a factor of 1.4. In the
case of sand, the Shields stress in the active channels during
major flow events can always be expected to be at least an
order of magnitude greater than the value at the threshold of
motion. The same lowered slope is driven, instead, by an exponent n in the sediment transport relation that exceeds the
value of 1.5 of the standard Meyer-Peter and Muller (1948)
relation used for gravel.
In the companion paper, Parker et al. (1998), the analysis
is applied with some extension to the case of a tailings basin
of a mine. Here the state of equilibrium aggradation is
achieved by building up the containing dikes at a rate just
sufficient to prevent the escape of tailings. The treatment results in a design tool that can be used to examine options to
extend the lifetime of the tailings basin.
An extensive series of experiments on fan morphology were
performed in the course of the research that led to the present
paper. The data so collected are presented and analyzed in the
context of fractional sheet flow in Whipple et al. (1998). The
role of morphology on the closure relations for sediment transport and resistance used in the present analysis is studied in
detail in Paola et al. (1998).
ACKNOWLEDGMENTS
The research reported here was funded by the parent company of the
"Rolling Stone" Mine and the National Science Foundation (grants CTS9207882 and CTS-9424507).
APPENDIX I.
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APPENDIX II.
NOTATION
The following symbols are used in this paper:
Bae
Bf
Cf
D
g
H
= effective total width of lumped channel actively transporting sediment;
= fan width;
= friction coefficient;
= representative grain size;
= gravitational acceleration;
= depth of lumped channel actively transporting sediment;
I
flood intermittency;
L = fan length;
n
p
Qs
Qso
Q",
qs
= exponent in sediment transport relation;
exponent in resistance relation;
= volume sediment transport rate in lumped channel actively
transporting sediment during floods;
upstream value of Qs;
water discharge;
volume sediment transport rate per unit width in lumped
channel actively transporting sediment during floods =
Qs/Bae ;
R = submerged specific gravity of sediment = (Ps - p)/p;
r = radial coordinate from fan apex;
i' = dimensionless radial coordinate = r/L;
S = slope;
t
time;
t. = effective time = It;
U = mean flow velocity in lumped channel during floods;
u*
friction velocity;
Vd
vertical rate of dike raising or base level rise;
Vs
vertical rate of basement subsidence;
ab
coefficient defining channel form;
ar
coefficient in resistance relation;
a ra = coefficient of adjustment in resistance relation to account
for morphology;
a ro = coefficient in resistance relation for flume-like channel;
as
coefficient in sediment transport relation;
asa = coefficient of adjustment in sediment transport relation to
account for morphology;
a so = coefficient in sediment transport relation for flume-like
channel;
T] = bed elevation;
e = fan angle;
Ap = porosity of sediment deposit;
p = water density;
ps = sediment density;
7b = boundary shear stress;
7*
dimensionless Shields stress;
Shields stress of lumped active channel;
7~ = critical Shields stress for sediment motion; and
X = coefficient defining fraction of aerial coverage of sheet flood.
7:
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