ALLUVIAL FANS FORMED BY CHANNELIZED FLUVIAL AND … · The present paper is devoted to fluvial...

11
ALLUVIAL FANS FORMED BY CHANNELIZED FLUVIAL AND SHEET FLOW. I: THEORY By Gary Parker/ Member, ASCE, Chris Paola,:1 Kelin X. Whipple/ and David Mohrig 4 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. 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. FIG. 1. View of Alluvial Fan of Kosi River, India, Showing Lo- cation of Main Thread of Flow at Various Times [Adapted from Gole and Chitale (1966)] JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998/985 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 ex- tent, it must eventually reach a point at which all of the sed- iment brought into its head is consumed in providing the de- posit 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 con- stant 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 ris- ing base level, e.g., that of a lake or the ocean. This mecha- nism 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 in- variant base level. Whether actively prograding outward or in a state of equi- INTRODUCTION Alluvial fans are fan-shaped zones of sedimentation down- stream of an upland sediment source. Fan shape tends to be regular and conical, and can often be described as axially sym- metric to a first approximation (Hooke and Rohrer 1979). At least three mechanisms for their formation have been ob- served; 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 ag- glomeration 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 fan- deltas (Nemec and Steel 1988). Any given fan may be built up by some combination of the previously mentioned mech- anisms. It is fair to say, however, that fans dominated by debris flows tend to have higher slopes than those dominated by flu- vial 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 sedi- mentary 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. 2Assoc. Prof.• Dept. of GeoJ. and Geophys., Univ. of Minnesota. Min- neapolis. MN 55455. 3Asst. Prof.. Dept. of Earth. Atmospheric. and Planetary Sci.. Massa- chusetts 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 sub- mitted 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. Nepal 87"10' a: nKosi ft su Barahakahetra Chalra li.l..l..l..J 25Km N f Downloaded 21 Nov 2011 to 131.215.67.4. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org

Transcript of ALLUVIAL FANS FORMED BY CHANNELIZED FLUVIAL AND … · The present paper is devoted to fluvial...

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 ofconstantly avulsing river and stream channels, those constructed by sheet flows, and those resulting from. ~esuccessive deposition of debris flows. The present analysis is directed toward the first two types. A mechamsticformulation of flow and sediment transport through river channels is combined with a simple quantification ofthe 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 fanformed predominantly by the deposition of sand is compared to a similar one formed predominantly by thedeposition of gravel. In each example the case of channelized flow is compared to the case of sheet flow. Themodel is applied to the tailings basin of a mine in the companion paper.

FIG. 1. View of Alluvial Fan of Kosi River, India, Showing Lo­cation of Main Thread of Flow at Various Times [Adapted fromGole and Chitale (1966)]

JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998/985

less acts to limit the horizontal growth of fans, as shown bythe 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 outwardprogradation ceases. Under these circumstances the fanreaches a state of equilibrium aggradation, at which the meanaggradation rate due to sediment deposition just balances thesubsidence rate, and mean elevation on the fan remains con­stant in time. This balance was described in a mechanisticsense 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 herethat the equilibrium aggradation associated with a base levelrising at a constant rate and vanishing subsidence is identicalto that associated with a constant rate of subsidence but in­variant base level.

Whether actively prograding outward or in a state of equi-

INTRODUCTION

Alluvial fans are fan-shaped zones of sedimentation down­stream of an upland sediment source. Fan shape tends to beregular and conical, and can often be described as axially sym­metric to a first approximation (Hooke and Rohrer 1979). Atleast three mechanisms for their formation have been ob­served; avulsing channelized rivers, sheet flows, and debrisflows (Schumm 1977; Blair and McPherson 1994). In the caseof a sheet flow, channelized upland flow reaches the fan andspreads out widely with no obvious channel, inundating muchof the fan surface and depositing sediment [e.g., Blair andMcPherson (1994)]. Debris flow fans are built up as an ag­glomeration of tongue-shaped deposits of individual debrisflows, each one of which is typically much smaller than thefan itself [e.g., Johnson (1984); Suwa and Okuda (1983);Whipple and Dunne (1992)]. Fluvial fans are built up by thesuccessive aggradation, and then avulsion of a river. The riverchannel 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 inFig. 1 (Gole and Chitale 1966; Wells and Dorr 1987). Fluvialfans may be completely terrestrial, or may have a distal portionwith standing water. Fans of the latter type are called fan­deltas (Nemec and Steel 1988). Any given fan may be builtup by some combination of the previously mentioned mech­anisms. It is fair to say, however, that fans dominated by debrisflows 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 fluvialfans. The analysis also allows for the description of sheet flowsas a limiting case.

It is not accidental that fans often form in regions that areundergoing subsidence in geologic time. Subsiding regions arenatural sinks for sediment; it is by this mechanism that sedi­mentary basins fill (Allen and Allen 1990). Subsidence, whilerarely exceeding rates of a few millimeters per year, neverthe-

'Prof.• St. Anthony Falls Lab.• Univ. of Minnesota. Minneapolis. MN55414.

2Assoc. 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. Toextend the closing date one month. a written request must be filed withthe ASCE Manager of Journals. The manuscript for this paper was sub­mitted for review and possible publication on January 27. 1997. Thispaper 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.

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Downloaded 21 Nov 2011 to 131.215.67.4. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org

librium aggradation, fans with a range of sizes in the sourcematerial usually display at least some degree of downstreamfining, 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 inwhich the gravel case can be generalized to grain mixtures.

It should be noted here that the achievement of equilibriumaggradation by a fluvial fan by no means imparts any stabilityto the river channel or channels on the fan. A constant meanbed elevation at any point is achieved only by the shift oravulsion of the river channel(s) over the entire fan, so as toprovide enough sediment to balance subsidence. Even in thecase of equilibrium aggradation, then, the channel(s) mustceaselessly rework the entire fan surface whenever the flow issufficient to render it active.

This ceaseless reworking at geomorphic time scales is thecause of numerous problems at engineering time scales(French 1987, 1992; Dawdy 1979). A significant fraction ofthe population of the Southwest United States lives on alluvialfans. A river channel on such a fan may appear stable fordecades, and then completely relocate itself on the fan surfaceafter a single flood event, with adverse consequences for thehomes, roads, railways, and bridges so affected. A recent ex­ample of a bridge problem associated with an alluvial fan isthe SH6 bridge over the Waiho River, New Zealand (Thomp­son 1991).

In the case of natural fans, most of the activity that buildsa fan occurs during floods, Le., when the channel(s) is mor­phologically at its most active. The vagaries of flooding aresuch that some can be associated with large sediment deliveryfrom the upland zone, and others with a smaller sediment de­livery. As a result, parts of the fan may undergo degradationfrom time to time, even though the fan is aggrading in thelong-term average. It has been postulated that fan buildup inthis 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 fanand depositing there [e.g., Schumm (1977)].

It is of interest, however, to consider the simplest possiblecase, i.e., one for which the discharge of water and sedimentat the fan head is constant in time, and for which the sedimentis 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 thesimplicity of the upstream boundary conditions may rule out

495

..... 485~

_i... 475w

300

a quasi-cyclic pumping of sediment, most other interesting fea­tures of fluvial fans are reproduced, including ceaselesslyavulsing channels and the evolution of a fan-shaped depositapproaching axial symmetry. A mechanistic explanation of fanmorphology under these simple conditions would be a usefulstep toward a predictive understanding of natural fan mor­phology.

Field examples of alluvial fans or fan-deltas that are not farfrom the simple case described before are provided by thetailings basins of mines. Efficient mine production requiresthat the delivery of water and tailings to the tailings basin mustbe quite constant from day to day. The grain size distributionof the tailings, however, rarely approaches uniformity; whilethe characteristic size is often in the range of sand or coarsesilt, there is often an admixture of mechanically producedgrains with sizes in the range of fine silt and clay. The fanformed in the tailings basin can thus be expected to display atleast some degree of downstream fining. Indeed, the finest ma­terial may deposit only in standing water. If the water is to berecycled with minimal sediment recycling, then a ponded zonemust 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 inflowof water and sediment must eventually achieve a state closeto equilibrium aggradation. This is illustrated in Fig. 2 for theWest Area number 1 of the tailings basin of an iron mine innorthern 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 thebasin must be built up over time. The case thus correspondsto fan formation under conditions of rising base level.

The present paper is one of a pair, with the companion paperbeing that of Parker et al. (1998). The present paper is devotedto the development of a theory of the evolution of an axiallysymmetric fan under the constraints of constant water and sed­iment inflow during floods and constant sediment size. Theeffect of varying hydrology is introduced in a simple way us­ing an intermittency factor corresponding to the fraction oftime the channel(s) is in flood. The results are then applied tothe case of equilibrium aggradation. It is shown that the caseof equilibrium aggradation balancing subsidence such thatmean absolute elevation does not change in time is mathe­matically equivalent to the equilibrium aggradation occurringin a confined basin such that bed elevation everywhere risesat an average rate that is constant in time. The results of theanalysis are then used to explore the role of fluvial channeli­zation in fan evolution. This is done by comparing the casefor which channelized flow is allowed to build the fan through

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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constant. It is assumed that no sediment escapes the basin, sothat the following boundary conditions apply:

It is assumed that at any given time during which there isflow, one or more active river channels course the fan. Thepresent description does not distinguish between single-chan­nel and braided configurations; Bac simply denotes the totalwidth of all the active channels measured along an arc normalto r. Thus

(3)

(5)

(4a)

(4b)

(2a,b)

(6a,b)

(4c,d)

d dQs vs- (q B ) =- = -(1 - A) - 9rdr s ac dr p I

where qs = mean volume sediment transport per unit width inthe channels. As these channels deposit sediment, they aggradeand thus shift or avulse to regions of locally low elevation onthe fan, so building up the fan across its entire width Bf . Thepresent formulation is averaged over a time scale that is longcompared to the time necessary to rework the entire fan sur­face at least once. Under this condition and the condition ofaxial symmetry, the mean bed elevation 1') at a point can betaken to be a function in space of r alone, with no angularvariation.

Under the foregoing constraints, during the fraction I oftime for which flow occurs on the fan the Exner equation ofsediment continuity takes the following form:

(aT] ) aQs( 1 - A)B - + v =--

p :t at s ar

Subsidence continues even during the fraction (1 - I) of timefor which there is no flow, during which time the Exner equa­tion takes the form

(1 - Ap)Bf e~ + vs) = 0

Imposing (2a,b) and solving, it is found that

g;. = [I - (£)']. T= ~ (I ~:,)9L'

The two prior equations can be averaged over time by multi­plying (4a) by I, (4b) by (1 - I), and adding. After somereduction the following time-averaged result is obtained:

(aT] vs) aQs(1 - A)B - + - =-- t = It

p :t at, I ar"

In (4c,d) 1') now = hydrologically averaged bed elevation; andt. = effective time. The formulation has been adapted from theoriginal formulation of Paola et al. (1992). Two subcases areof particular interest here. The first, which is of general geo­logical significance, is that of a natural fan for which a perfectbalance between aggradation and subsidence has beenachieved, so that fan shape averaged over many avulsions nolonger evolves in time. In this case bed elevation 1') becomesconstant in time and (4c) reduces with (1) to

It is seen that the sediment transport rate Qs must decreaseparabolically down the fan in this case. Since the subsidencerate V s can be considered as the first order to be imposed bytectonic effects [e.g., Allen and Allen (1990)], (6b) indicatesthat 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 headthan distance L. Once this length is reached, all the sediment

JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998/987

EXNER FORMULATION FOR FAN EVOLUTION

(1)

avulsion against the case of pure sheet flow with no chan­nelization. The assumption of constant water and sedimentdischarge is relaxed in a simple way, with the use of anintermittency, to allow for predictions that more closely ap­proximate nature. As illustrated next, the theory predicts thatchannelization acts to significantly reduce fan slope comparedto what would prevail under sheet flow.

In the companion paper, Parker et al. (1998), the theory isapplied to the engineering problem of the fan in the tailingsbasin of the "Rolling Stone" Mine. The assumption of uni­form sediment is modified in a simple way to account for thegrain size distribution. The analysis leads to an engineeringtool 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). Theformer paper is devoted to a study of experimental fans formedin the laboratory; the experimental results are comparedagainst the theory presented here. In the latter paper the roleof 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 reflectthe diversity of flow conditions associated with meandering orbraided stream morphology.

FIG. 3. Definition Sketch for Conical Fan

The geometry of Fig. 3 is considered. In plan view the fanis assumed to have a conical shape with angle 6 and radius L.The fan width Bf defines an arc given as a function of theradial coordinate r as

The basin itself may be undergoing subsidence at a verticalspeed V s ' Here, the subsidence rate is taken to be constant anduniform [e.g., Whipple and Trayler (1996)], although the the­ory is easily generalized to spatially and temporally varyingsubsidence rate. The volume discharge of water Qw, the feedrate of sediment Qso at the head of the fan, and the grain sizeD of the feed sediment are assumed to be constant for fractionI of time, where I is an intermittency, and vanishing for frac­tion (1 - 1) of time (Paola et al. 1992). Loss to ground wateris neglected, so that the total water discharge Qw crossing thearc normal to r is assumed to be constant everywhere. Thetotal sediment discharge Qs crossing the arc normal to r is notconstant, however, because the fan is built up by depositionassociated with a downstream decline in sediment discharge.The porosity Ap of the deposited sediment is assumed to be

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(11)

(13)

(12a)

(12b)

(14c)

(l4a,b)

q, = a ('1'* - T*)nVRgDD' C

p, TbR =- - 1, '1'* =--

P pRgD

where p = water density. It is assumed that

!:!- = a, (!!.)Pu* D

where a,o = resistance coefficient that would prevail in thecase of flow in a straight flume-like channel; and adjustmentcoefficient a,a = order-one factor providing bulk accountingfor the variability associated with an actual meandering orbraided channel [Paola (1996); Paola et al. (1998); see text tofollow].

A generalized sediment transport law of the form of Meyer­Peter and Muller (1948) is assumed here as

Here g = gravitational acceleration; as = dimensionless coef­ficient; n =dimensionless exponent

channel shear velocity, which is, by definition, related to theaverage boundary shear stress Tb as follows:

and T~ = critical Shields stress associated with the thresholdof sediment motion. The exponent n takes a value of 1.5 inthe original relation of Meyer-Peter and Muller (1948), whichwas obtained for the case of relatively uniform gravel. In thecase of the transport of sand at conditions well above thethreshold of motion, however, the formulation of Engelundand Hansen (1972) has an exponent of 2.5. The van Rijn(1984a,b) relation for sand can also be locally fitted to theform of (13) for the same transport conditions, yielding valuesof n that are again substantially higher than 1.5 (Whipple etal. 1998).

As was done for the resistance relation, the coefficient as isexpressed as

where a, = dimensionless coefficient; and p = dimensionlessexponent. The choice p = 1/6 yields the original Manning re­lation; the choice p =0 yields a Chezy resistance relation. Itmust be recognized that in any given case (12) would describea mixture of skin friction and form drag in some unspecifiedratio. It is useful to express the coefficient a, as follows:

where a,o = coefficient for the case of a straight, flume-likechannel; and a,a = order-one coefficient quantifying the bulkeffect of variations in meandering or braided channels (Paola1996; Paola et al. 1998). In general, a sa can be expected to begreater than unity. This is because in any river such parametersas the bed shear stress vary locally about their mean values,being high in some places and low in others. The nonlinearityin any typical sediment transport relation is such that zones ofhigh shear stress, for example, should contribute dispropor­tionately to the sediment transport. The effect in bulk is anincreased rate of sediment transport relative to that whichwould be predicted based on the locally constant shear stressin a flume-like channel.

It is seen from both (12) and (13) that knowledge of anaverage boundary shear stress Tb is required. Here it is assumedthat nowhere does the flow deviate too strongly from equilib­rium, or normal conditions, so that local equilibrium momen­tum balance reduces to

(7)

(8)

(10)

(9a,b)

d dQs Vd- (q B ) =- = -(I - A ) - 6rdr s oc dr P I

and thus with the imposition of (2a,b)

delivered to the head is consumed in balancing subsidence,leaving nothing remaining to drive further progradation.

The second subcase has engineering as well as geomorphicsignificance, as shown in the companion paper, Parker et al.(1998). Here the subsidence rate is assumed to be zero, or atleast negligible over the engineering time scale. The basin isassumed to be confined by dikes along its sides and at itsdownstream end; water is allowed to escape (perhaps to berecycled) but no sediment escapes. During periods of flow thedikes are assumed to be built up in time at a constant verticalspeed Vd so as just to contain the sediment. During periodswith no flow the dikes are not built up. The geometry thuscorresponds to a tailings basin of a mine. The state of equilib­rium aggradation is achieved everywhere when the bed buildsup at the same speed Vd as that of dike-building, so that

INTERNAL RELATIONS

Note that (9a,b) are essentially identical to (6a,b); the onlydifference being that the subsidence rate v, is replaced withthe rate of dike raising Vd' There is, however, a conceptualdifference. In the case of natural equilibrium aggradation bal­ancing subsidence, v, is an independent variable and basinradius L is a dependent one. The situation is reversed in thecase 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 thecase of a natural fan-delta that is responding to base levelrising at speed Vd rather than subsidence.

where t. is given by (4d); 'TILo = elevation at the downstreamend of the basin at t = 0; and basin elevation 'TI, relative to theelevation at the downstream end at any time is taken to be afunction of r alone. Setting V s equal to 0 in (4a) and reducingwith (1) and (7), it is quickly found that

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 describessediment transport, and the third specifies the total width ofthe active channels Bac • Here general relations of simple powerfonn are adopted, in that they can be used to obtain analyticalsolutions for equilibrium aggradation. The framework of thetheory is sufficiently flexible, however, to encompass quitegeneral internal relations.

While in reality the flow on the fan may be single-channelor braided at any given time, all the channels are here lumpedinto a single "effective" channel with average depth H, flowvelocity U, and sediment transport per unit width qso (Fig. 3),such that

The statistical implications of this procedure are discussednext. The relation for flow resistance is assumed to be of thegeneralized Manning-Strickler fonn. Let u* denote average

988/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998

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A more common case can be described as fractional inundationby sheet flow. Here this is treated in terms of a spreading anglethat is a set fraction of fan angle e, so that Bac is given by

• multi

• single

• "RS". .- ,;..100

10.00

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). Onlysand-bed streams with a characteristic grain size D less than0.5 mm are considered here. 1\\'0 data subsets were extractedfrom the compendium. The first pertains essentially to singlechannel meandering streams (sinuous channel, meandering, orwith minor secondary channels). The second pertains to mul­tiple channel streams (occasional split channels to braidedchannels). Flow conditions were chosen to be formative, Le.,bankfull; where this was unavailable, the two-year flood waschosen. Paved or degrading channels were excluded, as werechannels with nonalluvial banks. Seven single channel streamsand 11 multiple channel streams were selected in this way. InFig. 4 the Shield stress T: at formative discharge is plottedagainst slope S. Also included in the plot are data correspond­ing to averages for three large subsets of measurements inactive sand-bed braid channels in the tailings basin of the"Rolling Stone" Mine, which are explained in more detail inthe companion paper.

While the data display significant scatter, the values of T:show no significant trend in slope S, lending credence to thecrude approximation of a roughly constant value of T:. Theoverall average for T: 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 thedata fit within the band 1.33 :s; T: :s; 3.05, with an averagevalue 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; multiplechannel streams, 1.84; "Rolling Stone" Mine tailings basin,1.81. Within the scatter of the data, an overall crude estimateof T: of 1.8 would appear to be appropriate for alluvial sand­bed streams of all planforms but with a characteristic size Dbelow 0.5 mm.

It is not implied here that all, or even most, of the reachesof the streams in question are on alluvial fans. The purpose ofFig. 4 is rather to demonstrate that (18) is a crude but notunreasonable assumption for sand-bed streams in general. Acomparison of the consequences of the closure assumptions(17) and (18) for the active width of sediment transport allowsfor a comparison of the morphology of channelized fluvial fanswith 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 tailingsbasin of the "Rolling Stone" Mine is channelized into abraided 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 todescribe the case of interest. Perhaps the least reliable of the

(15)

(16)

(17a)

(17b)

(l8a,b)

Tb = pgHS

H- S = CXb CXb = RTa*D '

S= _a'T)ar

In the case of a relatively steep fan with braided channels, theFroude number of the flow may be high enough to render thiscrude assumption of local equilibrium reasonable as a first­order approximation. It is likely to be less accurate when theslope of the fan is low and a single, meandering channel pre­vails. In the latter case, a more correct formulation may thusrequire a description based on the assumption of graduallyvaried flow.

The final relation is the one governing the width Bac acrosswhich sediment is transported by the flow. In the extreme caseof pure sheet flow inundating the entire fan, Bac is simplygiven by fan width, i.e.

where X = constant between 0 and 1. While such a flow coversonly a portion of the fan at any given time, it is assumed toperiodically shift so as to lead to deposition over the entirefan in the long term. The experimental study of Whipple etal. (1998) addresses the issue of fractional sheet flow.

Another case of interest here is that of a channelized, orfluvial fan. In the case of a meandering channel Bac is simplythe mean width of the channel over which sediment is activelytransported; in the case of a braided channel it is the mean ofthe sum of the active widths. Here the assumption introducedto close the problem does not even directly involve Bac • It is,instead, assumed that channels evolve on the fan in such away that the "effective" channel maintains a constant Shieldsstress T: associated with active, mobile-bed conditions. From(l4b) and (IS), then, the relation for channel form can beexpressed as

where S = channel slope. Since Tb and H are bulk values char­acterizing the flow in the channels and local equilibrium isassumed, S may also be identified with local average fan slope(averaged over many avulsions), Le.

where CXb is taken to be a constant. That this does, indeed, leadto a closure for Bac is demonstrated next.

The form of (18) is easily justified as an approximation inthe case of active self-formed channels with gravel bed andbanks. Parker (l978a) provided a theoretical justification for avalue 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 basinfilling 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 formof the relation in terms of (HID) and S is suggested by therelation obtained by Parker (1978b) for sand bed streams

H mDS =cxcp (19)

S

FIG. 4. Values of Shields Stress T: under Formative Condi­tions for Various Rivers

where m = 0.6; and cxcp = dimensionless function of a Reynoldsnumber based on particle diameter D. An empirical justifica­tion for (18) for a specific case is presented in the companionpaper.

0.10

1.00E-05 1.00E-03 1.00E-02

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Q.

SOLUTION FOR FAN UNDERGOING EQUILIBRIUMAGGRADATION

internal relations is the one describing the Shields stress ofsand-bed streams at fonnative flow.

(28),.= !..L

For the case of channelized flow, substitution of (60) into (26)yields

S ={R-1I2a;la~+2P)/2a, (ex; _T~)-n [ Q.o(~~ ,.2)]} l/(l+p) (29)

OBSERVATIONS ON ROLE OF CHANNELIZATION

The elevation profile 'Tl(r) is found by integrating (27) or(29) subject to the boundary condition of a constant knownelevation 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 thecase of a tailings basin (or rising base level).

tion and subsidence and that pertaining to a balance betweendeposition and dike raising (or base level rise), results in thesame mathematical fonn for the downstream variation of slopeS on the fan. For the case of partial sheet flow, substitution of(6a) and (17b) into (23) yields

S ={R [(..!.. Q.o(l - ,.2) )lIn + T*]}(3+2p)/(2+2pla.yRgDDx6L,. c

'al/(l+p) ( Qw )-l/(l+P)

, Vii5Dx6Lf (27)

where

(20)

(22a)( )

-II(1+Pl'a1/(l+p) Qw

, Vii5DBac

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 QwViDDBac

=a [R-Ia-2/(3+2plS(2+2p)/(3+2P) ( Qw )2/(3+2pl _ T*]"., ViDDB

acC (21)

These fonns are independent of the assumption for the effec­tive width of the active channels Bac • They specify local av­erage Shields stress T* and sediment transport Q. as functionsof water discharge Qw and local average slope S. Eq. (21) maybe inverted to give S as a function of Qw and Q.

{ [ (

1 Q. ) 11" ] }(3+2p)/(2+2P

)

S=R - +T~a. yRgDDBac

The case of pure sheet flow is realized by setting X equal tounity.

In the case of channelized flow, (21) can be reduced with(18) to yield

In the case of fractional sheet flow, (17b) can be substituteddirectly to eliminate Bac from (20)-(22), yielding

{ [ ( )

1/"]}(3+2P)/(2+2pl

S - R..!.. Q. + T~- a.YRgDDx6r

Bac =a;(3+2pll2a ;IS(I+Pl (vfi 2) (23)D gDD

or reducing further with (22a)

Bac _ -I ( Q. ) (ab _ *)-" (24)D - a. yRgDD2 R Tc

Using (13), (18), and (24), the relation for sediment transportQ. as a function of Qw and S becomes

Q. _ -(3+2p)/2 -I (ab _ *)" S(l+P) ( Qw ) (25)~,;;-;:: 2 - a.ab a, R Tc • r-;:: 2v RgDD vgDD

or inverting for S as a function of Qw and Q.

S =[R-1I2a;la~+2p)/2a, (~ _ T~ ) -" (~:) ] I/(I+p) (26)

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 growthof 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-

990 I JOURNAL OF HYDRAULIC ENGINEERING I OCTOBER 1998

(30)

(31)

T* » T~

Here T* is given by (22) and (17b) for the case of partial sheetflow and T* = T: according to (18) for channelized flow. Underthese conditions it is easily shown that (22) and (26) reduceto the same relation for fan slope

S = R a, Q.a. Qw

Channelization is thought to play a fundamental role in thedevelopment of a fluvial fan morphology that is distinct fromthat created by sheet flows [e.g., Schumm et al. (1987)]. Thenonlinearity of sediment transport relations is such that for thesame discharge more sediment can be carried by a relativelynarrow, deep channel than a relatively wide, shallow one. Con­comitantly, if the sediment discharge is held constant, thechannelized flow, being the more efficient transporter of sed­iment, should maintain a lower slope. This lower slope can beimposed on the entire fan by the mechanism of avulsion. Theimplication is that channelization acts to lower the slope ateach point on the fan below the value that would prevail forunchannelized sheet flow.

A rather interesting limiting case serves as a foil for a dis­cussion of the effect of channelization on fan morphology. Letp = 0 (Chezy resistance relation), n = 3/2 (Meyer-Peter Mullersediment transport relation), and let the Shields stress T* sat­isfy the condition

That is, slope obeys a simple linear dependence on the ratioof the sediment discharge to the water discharge. Note that thecoefficient for width closure ab drops out of the fonnulationfor channelized slope in this case. Were the stated assumptionsto hold true in general, channelization would play no role inreducing fan slope below the value that would prevail for sheetflow.

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

(22b)( )

-II(1+P)

'a1/(I+p) Qwr Vii5Dx6r

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Manning value being 116), so a relaxation of the assumptionp = 0 plays little role in regard to the effect of channelization.If the exponent n in the sediment transport relation is keptequal to 3/2 but the Shields stress is no longer constrained tosatisfy (30), it is found that the slope of a channelized fandrops below the corresponding unchannelized case. If, on theother hand, (30) is retained but n is allowed to increase beyond3/2, it is again found that the slope of the channelized fandrops below the corresponding unchannelized case.

Each scenario has its expression in the real world. TheMeyer-Peter and Muller (1948) sediment transport relation is,perhaps, the most reasonable of the simpler sediment transportequations for gravel transport. As noted before, however, self­formed gravel-bed channels with gravel banks tend to maintainShields stresses that are not far above the critical value formotion. As illustrated by the following example problem, theresult is channelized fan slopes that are significantly less thantheir unchannelized counterparts.

Self-formed sand-bed streams typically maintain Shieldsstresses at formative (i.e., bankfull) flows that are greatly inexcess of the threshold value for transport. A simple fit of themore reliable relations for the transport of sand into the formof (13), however, yields values of n in excess of 1.5 in thelimit of high Shields stress. A simple illustration of this canbe given with the total sand transport relation of Engelund andHansen (1972). It takes the form

The value of n is seen from (32) to be 2.5; comparing with(13) it is seen that T: can be effectively set equal to zero.(This does not imply that there is no threshold of motion inthe Engelund-Hansen formulation; rather, it implies that it doesnot appear explicitly in the load equation, itself.) Upper-regimeflow is taken as an example here, as it can be expected to berather 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 thusslowly varying, function of HID. Taking the rather typicalvalue 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 obtainedby means of a local fit of the sand transport relations of vanRijn (1984a,b) to (13) (Whipple et a1. 1998).

The foregoing observations are used as the basis for a morespecific calculation. A subsiding basin with a radius of 10,000m 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 ofthe 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 be0.4. These values are given in Table 1.

Two cases summarized in Table 1 are considered-one fora gravel fan and the other for a sand fan. In both cases thefan produced by channelized flow is compared to the one re­sulting from pure channelized sheet flow covering half the fansurface during any given flood (X = 0.5). In both cases a Chezyresistance relation, according to which p = 0 and aro =C;1I2, is postulated. In the case of the gravel fan characteristicgrain size D is taken to be 20 mm and C;lt2 is set equal to10. The Meyer-Peter Muller sediment transport relation is as­sumed, so that n = 1.5 and a so = 8. The critical Shields stressT: is set equal to 0.03 and the Shields stress of the activechannels T: is set to be 1.4 times T:, or 0.042, yielding a value

where Cf = friction coefficient given by the relation

C- 1/2 _ Uf ­

u*

(32)

(33)

TABLE 1. Assumed Parameters for Calculation of Fan Char­acteristics

Parameter Gravel fan Sand fan Notes(1 ) (2) (3) (4)

L (m) 10,000 10,000 Basin radius6 2.094 2.094 120·R 1.65 1.65 Specific gravity =2.65lI.p 0.4 0.4 PorosityD (mm) 20 0.30 -p 0 0 Chezy lawIX", 10 15 = Ci"2

IX.. 1 1 -n 1.5 2.5 -IX.o 8 11.25 -IX•• 3.0 1.5 -

"~ 0.030 0 -,.* 0.042 1.8 -.IX. 0.0693 2.97 =,.: RQw (m'/s) 200 20 -Q.o (m'/s) 0.1 0.04 -I 0.02 0.05 Intermittencyv. (mrn/yr) 1.00 1.00 Subsidence ratey (ttyr) 1.67 x 10' 1.67 X 10' Sediment yield

of ab of 0.0693. Insofar as gravel rivers tend to be ratherflashy, especially on fans, flow is assumed to occur only 2%of the time, or 7.3 days per year, yielding an intermittencyfactor I of 0.02. This value may be reasonable for humid andsubhumid fans, but high for desert fans. During the time offlow the water discharge Q", is assumed to be 200 m3/s, andthe sediment feed rate at the head of the fan is assumed to be0.1 m3/s. These numbers yield a subsidence rate V s of 1.00mm/yr in accordance with (9b) and a sediment yield to thebasin Y of 1.67 X 105 t/yr.

In the case of the sand fan, a value of grain size D of 0.30mm is selected; aro = C;It2, n, a so , and T: are set equal to IS,2.5, 11.25, and 0, respectively, in accordance with the priordiscussion. In accordance with the discussion associated withFig. 4 T: is set equal to 1.8, yielding a value of ab of 2.97 forsediment with a natural specific gravity of 2.65. This value ofactive channel Shields stress is well in excess of the thresholdvalue T: for the motion of sand, which should not exceed 0.06.Insofar as sand-bed streams typically have lower slopes thangravel-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 valuewould be much lower for a sandy desert fan. The sediment feedrate Qso during the period of flow is set equal to 0.04 m3/s, soas to give the same subsidence rate V s and sediment yield Yas the gravel fan. Water discharge Q", is set equal to 20 m3/sso that the concentration of sediment entering at the head ofthe sand fan is four times that of the gravel fan.

In both cases the morphologic adjustment coefficient a'a inthe 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 notlarge. 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. Acomparison of sediment transport in flumes and field rivers(most of which were meandering) led Brownlie (1981) to pro­pose a multiplicative adjustment coefficient corresponding toasa of 1.268 in his own sediment transport relation. This valuemay be an underestimate in that the measured sediment trans­port rates typically pertain to flow conditions in relativelystraight 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

JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998/991

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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 toincrease the effectiveness of sediment transport.

Here only loose estimates need be employed. In the case ofa gravel fan this coefficient is set equal to 3.0, a value thatwould appear to be in the right range for braided gravel-bedchannels based on the work of Paola (1996). In the case ofthe sand fan the value is lowered to 1.5, in anticipation of amorphology closer to meandering, and in light of the work ofBrownlie (1981) mentioned earlier. The choice of a lowervalue for exsa for the sand fan than for the gravel fan based onan expected difference in morphology is an assumption thatcan be tested a posteriori based on the predicted values for theaspect ratio of the active channel(s) BaclH.

The results of the calculations for the gravel fan are shownin Figs. 5(a)-(c), in which the spatial distributions of fan slopeS, width of the active channels BaC' and depth H for the caseof unchannelized sheet flow are compared with the channel­ized distributions. (Recall that Bac = XBf for the case of sheetflow; 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 ofmagnitude larger than channelized slopes. The channelizedslopes are seen to be fairly constant near the upstream end ofthe fan and dropping toward the downstream end. This impliesan elevation profile that is straight near the upstream end andconcave near the downstream end. In Fig. 5(b), unchannelizedactive widths are equal to half the fan width Bit and take valuesapproaching 10 km at the downstream end of the fan. Thechannelized active widths are on the order of 200 m, and de­cline in the downstream direction as a result of deposition. InFig. 5(c), unchannelized depths quickly drop off to the orderof centimeters, whereas channelized depth is on the order of1 m and increasing in the downstream direction. The effectsof channelization on the fan, as manifested through values ofT: that are only modestly above the critical value T:', are dra­matic.

The results of the calculation for the sand fan are equallydramatic, as shown in Figs. 6(a)-(c). In Fig. 6(a), unchan­nelized slopes are on the order of 0.01, whereas channelizedslopes are around 0.001 and declining in the downstream di-

r/L

•• Sheet-Channel

•• Sheeti-Channel

0.80.60.40.2

0.1

0.01

II) 0.001

0.0001

0.000010 0.2 0.4 0.6 0.8

r/L

10000

1000

I 100u

m

10

•• Sheet-Channel

- • Sheetl-ch<lnnell

0.80.60.40.2

100 •

1000

00001 L.-__'---_---l__-----'__~____'_J

o

u

m

0.001

0.01

r/L

10

§::I:

0.1

0.01

- • Sheet-Channel

0.001

o 0.2 0.4 0.6 0.8

r/L

FIG. 5. Profiles of Channelized and Unchannelized: (a) SlopeSfor Gravel Fan; (b) Active Channel Width S•• for Gravel Fan (InUnchannelized Case, Width Is Half Fan Width S,); (c) Depth HforGravel Fan

r/L

FIG. 6. Profiles of Channelized and Unchannelized: (a) SlopeS for Sand Fan; (b) Active Channel Width S •• for Sand Fan (InUnchannelized Case, Width Is Half Fan Width S,); (c) Depth H forSand Fan

992/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998

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r/L

FIG. 7. Plot of BaclH versus rlL for Channelized Gravel andSand Fans

1- - Sand-Gravel

The present analysis is, of necessity, quite simplified. Sincethe channels are not explicitly modeled, the predictions for fanmorphology represent only averages. The model would requireconsiderable generalization before it could describe the cut­and-fill sequences associated with strongly varied water andsediment supply (Schumm 1977).

The generalization of the model to include downstream fin­ing would be more straightforward. For the case of gravel, forexample, the simple bulk formulation of the type of Meyer­Peter and Muller (1948) used here could be replaced by arelation that explicitly includes grain size variation [e.g.,Parker (1990, 1991a,b)]. While the analysis here focuses onequilibrium 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 stateis characterized by a distinct prograding front, a feature easilyadded to the present formulation.

Two objections can be raised against the use of a constantShields stress T: for the active channels. While the assumptionof a constant value of T: that is not far above the threshold~al~e fo~ sediment motion has both theoretical and empiricalJustificatIOn for gravel channels, within any agglomeration ofindividual active braid anabranches T: can be expected to varyprobabilistically. Paola (1996) and Paola et al. (1998) haveincluded this feature, which results in an adjustment coefficientin the resistance relation elTa that is typically close to unity andan adjustment coefficient in the load relation elsa that can beas 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 thiscase the assumption of a constant value of T: well above the

JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998/993

14

16 ,-------,------------------.

4

ther of these profiles is seen in the field [e.g., Bull (1977)], anobservation that may be useful in interpreting outcrops.

SUMMARY

12

- - Sand-Gravel

080.60.40.2

1\-------~--_+_--_I_----'-----!.j

o

100

10

rection. As in the case of the gravel fan, the elevation profileof the channelized sand fan can be expected to be straight nearthe upstream end and concave near the downstream end. InFig. 6(b), channelized active width Bae is seen to be around20 m and declining in the downstream direction as sedimentis deposited. The predicted values are much less than the widthof the sheet flow. In Fig. 6(c), channelized depths are seen tobe on the order of 1 m and increasing in the downstreamdirection, whereas unchannelized depths quickly drop to theorder of centimeters. Here the difference between the chan­nelized and unchannelized cases is mediated by the degree towhich the exponent n in the sediment transport equation ex­ceeds the value of 1.5.

The depth profiles for both the gravel and sand fans indicatethat a sheet flow at the specified discharge covering half thefan surface would become so shallow as to be unsustainablebeyond 30% or 40% of the fan length. Calculations of thistype 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 beinferred from Fig. 7, where the ratio BaJH is plotted versusrlL for the channelized gravel and sand cases. The value ofthis ratio is around 300 on the upper part of the gravel fan,but only around 40 for the sand fan. Incipient braiding isknown to be associated with a high width-depth ratio [e.g.,Blondeaux and Seminara (1985)]. Appropriate estimates of thevalue of channel width to depth for incipient braiding are inthe range of 100-180, with the lower value corresponding toa higher slope [e.g., Hokkaido (1987)]. Evidently the gravelfan 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 sandfan can be expected to be single-channel meandering or sin­uous throughout most of the fan.

It should be noted that the assumption for channelizationemployed here breaks down at the downstream end of the fan.According to (26), slope S must vanish at the downstream endof the fan; (18) then implies that channel depth H must tendto infinity there. Evidently the channel closure hypothesis failson extremely low slopes, a result that is, perhaps, not unex­pected. In the case of a fan-delta, the channels should giveway to sheet flow over deltaic lobes at the extreme distal end.

The predicted elevation profiles for the channelized graveland sand fans are shown in Fig. 8(a). As suggested earlier, theupstream half of both is approximately linear and the down­stream half of both is upward concave. The elevation profilesof the unchannelized fans are shown in Fig. 8(b). The profileof the gravel fan is upward convex everywhere, and the profilefor the sand fan is, likewise, upward convex everywhere ex­cept near the distal end, where a slight concavity is seen. Nei-

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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 relationfor 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 ofthe loss of water from the channel(s) due to infiltration maynot always be justified. While it is relatively easy to modifythe theory to incorporate any specified pattern of infiltration,the process is quite difficult to predict a priori. Suffice it tonote 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. Theeffect of varied discharge and sediment supply is introducedin a simple way by means of an assumed intermittency of flow.In the case of sheet flow, the flow is assumed to inundate someset fraction of the fan width with no channelization. Pure sheetflow 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 shiftso as to result in deposition across the entire fan. In the caseof a channelized fluvial fan, the active channel(s) is assumedto maintain a Shields stress near a constant value of ,.:. Whileat any instant the flow is assumed to be channelized, depositedsediment is spread across the width of the fan in order todescribe the consequences of channel shift or avulsion. Whilethe 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 thebasin or rising base level.

The analysis indicates that channelization on fluvially dom­inated alluvial fans has the effect of dramatically reducing fanslope as compared to pure unchannelized sheet flows. Whileboth the gravel and sand fans display this same feature, it ismediated by somewhat different mechanisms. In the case ofgravel, the lowered slope is driven by a Shields stress in theactive channel(s) that is allowed to exceed the threshold valuefor the movement of sediment only by a factor of 1.4. In thecase of sand, the Shields stress in the active channels duringmajor flow events can always be expected to be at least anorder of magnitude greater than the value at the threshold ofmotion. The same lowered slope is driven, instead, by an ex­ponent n in the sediment transport relation that exceeds thevalue of 1.5 of the standard Meyer-Peter and Muller (1948)relation used for gravel.

In the companion paper, Parker et al. (1998), the analysisis applied with some extension to the case of a tailings basinof a mine. Here the state of equilibrium aggradation isachieved by building up the containing dikes at a rate justsufficient to prevent the escape of tailings. The treatment re­sults in a design tool that can be used to examine options toextend the lifetime of the tailings basin.

An extensive series of experiments on fan morphology wereperformed in the course of the research that led to the presentpaper. The data so collected are presented and analyzed in thecontext of fractional sheet flow in Whipple et al. (1998). Therole of morphology on the closure relations for sediment trans­port and resistance used in the present analysis is studied indetail in Paola et al. (1998).

994/ JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 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. REFERENCES

Allen, P. A., and Allen, J. R. (1990). Basin analysis: Principles and ap­plications. Blackwell Scientific Publications, Oxford, U.K.

Blair, T. C., and McPherson, J. G. (1994). "Alluvial fans and their naturaldistinction from rivers based on morphology, hydraulic processes, sed­imentary processes and facies assemblages." J. Sed. Res., A64(3),450-489.

Blissenbach, E. (1952). "Relation of surface angle distribution to particlesize distribution on alluvial fans." J. Sed. Res., 22, 25-28.

Blondeaux, P., and Seminara, G. (1985). "A unified bar-bend theory ofriver meanders." J. Fluid Mech., 157,449-470.

Brownlie. (1981). "Prediction of flow depth and sediment discharge inopen channels." Rep. KH-R-43A, W. M. Keck Lab. of Hydr. and WaterResour., California Inst. of Technol., Pasadena, Calif.

Bryant, M., Falle, P., and Paola, C. (1995). "Experimental study of avul­sion frequency and rate of deposition." Geology, 23(4), 365-368.

Bull, W. B. (1977). "The alluvial fan environment." Progress in Phys.Geography, I, 222-270.

Church, M., and Rood, K. (1983). Catalogue of alluvial river regimedata. Dept. of Geography, Univ. of British Columbia, Vancouver, B.C.

Cui, Y., Parker, G., and Paola, C. (1996). "Numerical simulation of ag­gradation and downstream fining." J. Hydr. Res., 34(2), 185-204.

Dawdy, D. R. (1979). "Flood frequencies on alluvial fans." J. Hydr.Engrg., ASCE, 105(11), 1407-1414.

Engelund, E, and Hansen, E. (1972). A monograph on sediment transport.Technisk Forlag, Copenhagen, Denmark.

French, R. H. (1987). Hydraulic processes on alluvial fans. Elsevier Sci­ence Publishing Co. Inc., New York, N.Y.

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APPENDIX II. NOTATIONThe following symbols are used in this paper:

Bae = effective total width of lumped channel actively transport-ing sediment;

Bf = fan width;Cf = friction coefficient;D = representative grain size;g = gravitational acceleration;H = depth of lumped channel actively transporting sediment;I flood intermittency;L = fan length;n = exponent in sediment transport relation;p exponent in resistance relation;

Qs = volume sediment transport rate in lumped channel activelytransporting sediment during floods;

Qso upstream value of Qs;Q", water discharge;qs 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;V s vertical rate of basement subsidence;

ab coefficient defining channel form;a r coefficient in resistance relation;

a ra = coefficient of adjustment in resistance relation to accountfor 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 toaccount for morphology;

a so = coefficient in sediment transport relation for flume-likechannel;

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;7: Shields stress of lumped active channel;7~ = critical Shields stress for sediment motion; andX = coefficient defining fraction of aerial coverage of sheet flood.

JOURNAL OF HYDRAULIC ENGINEERING / OCTOBER 1998/995

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