ARMANINI Debris Flow

142: Debris Flow

ARONNE ARMANINI, LUIGI FRACCAROLLO AND MICHELE LARCHER

CUDAM and Department of Civil and Environmental Engineering, University of Trento,Trento, Italy

The principal features of debris flow are described. One section is devoted to debris-flow triggering, accountingfor the geomechanical criteria by Takahashi (1978) and for most recent hydrological distributed models. In asecond section, the rheology of debris flows is described: the theory of dispersive stresses by Bagnold, the kinetictheories of dense gases, and some recent experimental observations about debris flows in equilibrium. A sectionis devoted to mathematical models and the final section discusses debris flows countermeasures.

INTRODUCTION

Debris flow is a motion of widely sorted debris (from afew millimeters to some meters) inside a watery matrixor also in the presence of clayish mud. One of the mostevident facts is the floatation of huge boulders on thesurface of the debris flow. Taking up the original definitionproposed by Takahashi (1981), debris flows are massivesediment transport phenomena that manifest themselves inmountain streams characterized by a steep slope, wherethe motion of the granular phase is induced directly bygravity. Here, the ratio between the liquid and the solidtransport rates is relatively low, and can be zero in caseof dry granular mixtures (e.g. dry landslides). On thecontrary, in the case of ordinary sediment transport (bedload and suspended load), sediments are not driven directlyby gravity, but by the hydrodynamic actions induced bythe fluid, and liquid–solid transport ratio is relatively high.There is a wide range of two-phase sediment-laden motionsfor which this distinction is not so sharp. Other situationsoutside simple schemes concern cases where, although thesolid transport rate is in the debris-flow range, the finestfractions of the grain distribution change the rheology of theinterstitial fluid. In fact, the role of fine sediments, becauseof their size, and in some cases, of their electrochemicalproperties, is completely different from the coarser ones:they can mix with water, forming a homogeneous interstitialfluid (slurry), typically characterized by high viscosity andin some cases by cohesive behavior. Its composition isstrongly connected to the flow dynamics, making it very

difficult to define and distinguish the role of the twophases. Further, complications take place when a thirdphase, constituted by air and other gases, is present insideand affects the flowing mixture, such as in the stout frontof advancing debris-flow waves.

The picture presented above is mirrored by manifold clas-sifications found in literature, where massive phenomenainvolving the liquid and solid phase are reported with dif-ferent terminology: for example, debris flows, mud flows,hyperconcentrated flows, and so on, any of them possiblybeing preceded by adjectives such as turbulent, laminar,and so on. Recently, the knowledge about granular flowshas progressed a lot, and nowadays (2003) there is a wideliterature regarding both the rheological features of thesekinds of flows and the applicative aspects concerning designcriteria for defence and protection works.

As a rule, there are at least two fundamental elementsthat must be present for the triggering of a debris flow: anopportune succession of rain events and the availability ofsolid material. As better explained afterwards, the weatherevent has to present absolutely particular characteristics:generally it must be very intense, but in order to beable to move the bulk it must also be preceded by anevent long enough to take the sediments to saturation. Theconcomitance of these happenings gives little predictabilityto debris-flow events that are relatively rare, but presenta remarkable destructive power; they concern above allsmall basins and in particular alluvial fans that haveoften undergone a recent urbanization. Basically, theyare nonstationary phenomena developing in particularly

Encyclopedia of Hydrological Sciences. Edited by M G Anderson. 2005 John Wiley & Sons, Ltd.

2174 OPEN-CHANNEL FLOW

short times. Because of their unexpected character, theirdestructive power is often undervalued: small creeks thatare affected by very modest solid or liquid events fordecades, are sometimes affected by debris flows of hugeintensity. The recurrence of such events, in fact, is verydifficult to determine; systematic observations show that thedebris flows have return times of the order of 50–100 years,a period of time during which the phenomenon is likely notto manifest at all. Unlike floods in water streams, in whichsome extreme events happen almost regularly every year,even though with a different intensity, for debris flows itis exactly the sudden and often unexpected triggering thatmakes the phenomenon insidious.

DEBRIS-FLOW TRIGGERING

The assessment of debris-flow risk is very difficult as threemain factors participate in the triggering of debris flows: therainfall intensity, the initial state of the basin soil moisture,and the presence of enough sediment. The analysis ofmany different events has shown that generally the debrisflows manifest themselves after an extreme rainfall eventfollowing a rainfall of long duration. In a certain area, infact, debris flows usually occur when the antecedent rainfallintensity exceeds a certain critical value. Yet, it is necessarythat in a suitable previous period (of the order of 7 to10 days) a congruous volume of rain has fallen, makingat least part of the valley saturated.

A more physically based approach consists in usinga steady-state shallow subsurface flow model (e.g. TOP-MODEL by Beven and Kirkby, 1979) to evaluate thesaturated areas and their expansion during the storm. How-ever, unlike landslides generated in hillslopes by subsurfaceflow, debris-flow initiation is mainly due to surface runoffand a complete rainfall-runoff model is required. If designof defence works is implied, a special kind of rainfall repre-sentation could be useful in the form of intensity-duration-frequency (IDF) curves that relate the rainfall intensity andduration to a prescribed return time. If just the peak dis-charge is required, some simplified models are availableand are sufficient for this goal (Rigon et al., 2005). Accord-ing to the theory presented below, once the runoff createsa flow depth greater than a suitable threshold, the slopecollapses and the debris-flow run-out starts. This criticalvalue depends obviously on the geometric, geologic, andmorphologic characteristics of the debris deposit and of thesublayer on which it lies.

This approach alone is, however, not sufficient to getthe total of sediment mass moved. Besides the weatherevents, there must also be a deep-enough debris storage sothat the frequency of debris flows does not coincide withthat of intense rainfall events. Hampel (1968) distinguishesbetween watercourses with a rocky channel bed and thosewhich flow on erodible alluvial deposits. The former require

a gradual debris accumulation between an event and thefollowing ones and are temporarily stable after the event;the latter can be destabilized by an event and this can bringabout a period of activity with rather high frequencies,followed by the return to a state of rest. The use ofstochastic models to assess the contemporary presence of allthe conditions required is presently (2003) being studied,but it is still not sufficiently tested to be utilized in fieldapplications (Iida, 1999).

A further possibility of debris-flow generation is obvi-ously due to the dynamic action of landslide in their run-outfrom hillslopes into channels or the collapse of retentionstructures existing along the mainstream. The total quantityof debris that rests after the event is usually due both to thesize of the source of material (especially if it derives froma landslide) and to the dynamics of the debris flow, whicherodes the bed on which it is moving.

Takahashi (1978) first introduced the stability theoryregarding loose materials for the study of debris-flowtriggering. This theory refers to a noncohesive and uniformgranular body and is based on the balance of forces actingin different imbibition conditions. The Takahashi instabilitycondition is written as follows:

tan α ≤ tan ϕC0�

1 + C0� + 1

n

h0

D

(1)

where α is the inclination angle of the deposit, ϕ isthe friction angle of the material, C0 is the volumeconcentration of the particles in a state of rest, h0 isthe flow depth of the water flowing over the deposit (inuniform-flow condition), � = (ρs − ρ)/ρ is the relativedensity of the immersed material, D the mean diameter ofthe material, and n is a nondimensional coefficient of theorder of the unit depending on the shape and distributionof the grains. Note that if h0 = 0, corresponding to acompletely saturated body with no surface runoff, thestability condition becomes:

tan α ≤ tan ϕC0�

1 + C0�(2)

Once in movement, the body is soon transformed into adebris flow. In this situation, the dynamic friction can alsomove the material lying underneath the layer initially madeunstable; this phenomenon is known as mass entrainment.Moreover, according to Takahashi (1978) the values ofthe ratio (h0/nD) able to generate a real debris-flow rangebetween 0 and 1.33. For values below 0, there is a partiallydry deposit which, when it becomes unstable owing to bigenough slopes, gives rise to a landslide. For values over1.33, there is a movement more similar to bed load ratherthan to debris flow. In short, according to Takahashi, the

DEBRIS FLOW 2175

slopes for which there is debris-flow range between the twofollowing extremes:

tan ϕC0�

1 + C0� + 1.33≤ tan α ≤ tan ϕ

C0�

1 + C0�(3)

If, for example, for the above-cited parameters weassume the following values that are characteristic of stony,noncohesive material, C0 = 0.7, � = 1.65, tan ϕ = 0.8, weobtain:

15◦9′ ≤ α ≤ 23◦5′ (4)

Once the movement is triggered, the debris flows canflow even with slopes smaller than the limit on the rightof relation (4). Anyway, with slopes lesser than 3◦ thedebris flows do stop. Consequently, the basins that aremore frequently affected by debris flows are those inwhich the presence of hillslopes or torrents with slopescomprised between the limits listed above are statisticallymore important.

The analysis of the historical events in the differentregions suggests that the debris-flow phenomena regardessentially small catchments, generally from 2 to 10 km2.Other elements that are important for the debris-flowformation are the geology and the vegetative cover. It isevident that the presence of vegetation inhibits stronglythe debris-flow formation; therefore, debris flows are morelikely to form above the vegetation limit. After all, it isat higher elevations that there are hillslopes with a sloperanging between the limits cited.

Other mechanisms generating debris flows are givenby landslides depositing in the channel bed, or by thecollapse of a natural dam which was formed temporarilyin the channel bed because of the stop of vegetationtransported by the current, or else by the collapse ofsome retention structures (in particular check dams) builtalong the mainstream of the river affected by an initialdebris flow.

DEBRIS-FLOW RHEOLOGY AND DYNAMICS

Once started, debris flows tend to assume a typical layout,formed by a round-shaped front in which the biggestboulders tend to accumulate, followed by a body in whichthe free surface is nearly parallel to the undisturbed debrisbed. In this part, the motion can be assumed to be close tothe uniform motion. The central part with uniform motion isfollowed by a tail, in which the flow depth becomes thinnerand where the zone of erosion is exposed (see Figure 1).

It is possible to study the behavior of the motion in thedifferent parts of the debris flow separately. It is, above all,the part with quasi-uniform motion that becomes signifi-cant, since the hypothesis of uniform flow makes it possibleto give a detailed determination of kinematic and dynamic

TailBody

Front

Figure 1 Scheme of the longitudinal section of a debrisflow: in the central part (body) the motion is quasi-uniform

conditions. The study of debris-flow dynamics requires theknowledge of the interaction between particles and inter-stitial fluid, between the contour, the particles, and theinterstitial fluid, and among the particles themselves. There-fore, the quantitative description of debris-flow dynamicsis very complex. It is often convenient to consider thedebris flow as a continuous medium, to which is assigned asuitable rheological law able to simulate the different inter-actions between particles, fluid, and wall. In fact, a granularmaterial subjected to deformation can determine differenttypes of interactions among grains and, therefore, it cangenerate stresses by different mechanisms. Individual parti-cles may interact with one another in rigid particle clusters,generating a network of contact forces through sustainedrolling or sliding contacts, or by nearly instantaneous colli-sions, during which momentum is exchanged and energy isdissipated because of inelasticity and friction. The relativeimportance of these mechanisms may be used as the char-acteristic defining the various flow regimes (Savage, 1984).Basically, the main interactions can be listed as follows:

– deformation of the mean fluid field by the particles;– collisions among the particles;– friction among the particles during long contact periods;– deformation of the turbulence structure generated by

the wall.

Depending on particle concentration and hydrodynamicconditions, some of the mechanisms described above canprevail with respect to the others.

Theory of Dispersive Stresses

Theoretical and experimental findings by Bagnold (1954),regarding the rheology of a gravity-free dispersion oflarge solid spheres in a Newtonian fluid under shear,probably represent the most extensively cited research overthe last 50 years among scientists involved in the studyof debris flows, snow avalanches, and of the flow ofgranular materials in general. Bagnold’s theory presentssome conceptual limits that have been analyzed in detailin the recent literature, among others by Hunt et al. (2002).Yet, it has the merit of being simple and of being the first

2176 OPEN-CHANNEL FLOW

one that indicated the physics of the phenomenon clearly;moreover, it has been successfully applied to debris flows,above all by Takahashi (1978).

The theory is based on the observation that whenparticles collide among themselves or on the containmentwall, collisions manifest themselves as an increase inpressure called dispersive pressure. According to Bagnold(1954), two extreme situations can occur. The first oneoccurs in the presence of particles of small size beingwell dispersed within the interstitial fluid: the debris flowassumes then a macroviscous behavior and the viscousstresses prevail with respect to the dispersive ones becauseof the intergranular collisions.

On the contrary, an opposed situation occurs whenthe particles have high concentrations and the debris-flowspeed is high, and then the collisions are much morefrequent. In this regime, called grain-inertia, the collisionsamong grains are the determining factors as to the debris-flow resistance and the effect due to the viscosity of theinterstitial fluid is negligible.

Bagnold distinguished grain-inertia and macroviscousflow regimes on the basis of the nondimensional parameterBa presented in equation (5), subsequently termed Bagnoldnumber, representing the ratio between stresses due toinertia and those due to viscosity.

Ba = ρD2(du/dz)

µλ1/2 with λ = 1

(C0/C)1/3 − 1(5)

where D is the grain diameter, λ is Bagnold linear con-centration, C is the volume concentration of the solidfraction, C0 is the maximum concentration possible (con-centration at rest), du/dz is the shear rate, µ and ρ are thedynamic viscosity and the mass density of the interstitialfluid. Bagnold called macroviscous the regime character-ized by small Bagnold numbers (Ba < 40), where the shearstresses behave like a Newtonian fluid with a viscosity cor-rected by the presence of the particles, and grain-inertiathe other regime, is characterized by Ba > 450, where thestresses were independent of the fluid viscosity and pro-portional to the square of the shear rate and to the squareof the linear concentration λ. The intermediate range ofBagnold number (40 < Ba < 450) occupies a transitionalregion.

Bagnold derived simple analyses to explain the rheolog-ical behavior in the two limiting regimes. In the viscosity-dominated macroviscous regime, shear and normal stressesare linear functions of the shear rate du/dz. Bagnold iden-tified the presence of a normal stress in radial direction,termed dispersive pressure, and attributed it to a statisticallypreferred anisotropy in the spatial particle distributions. Heproposed the following relations for the shear stress τ andthe normal stress σ :

in the macroviscous region : τ = 2.25λ3/2µdu

dz

σ = τ

tan ϕ(6)

and in the grain-inertia region :σ = 0.042ρ

(λ

du

dzD

)2

cos ϕ

τ = σ tan ϕ (7)

It should be noted that both in the macroviscous and inthe grain-inertia regime, the normal stress is proportional tothe shear stress in the form τ = σ tan ϕ, where ϕ representsa dynamic friction angle, depending on collision conditions.Such behavior is reminiscent of the Coulomb criterionused to describe the stresses in cohesionless soils underconditions of limited equilibrium. According to Brown andRichards (1970), typical values for ϕ obtained during quasi-static yielding at low stress levels are close to the angleof repose, that is, about 24◦ for spherical glass beads and38◦ for angular sand grains. Bagnold proposed a dynamicfriction angle ϕ ∼= 37◦ for the macroviscous regime andϕ ∼= 18◦ for the grain-inertia regime.

As a completion of Bagnold’s theory, nowadays themacroviscous regime is considered to be practically absentin debris flows, while where the shear rate and the velocityof the fluid are small, a regime called frictional or quasi-static takes place, in which the contact among particles isquasi-permanent and the ratio between normal and shearstress can be considered roughly constant.

Kinetic Theories

Within a granular flow, the velocity of each particle maybe decomposed into the sum of a mean velocity anda random component, taking into account the relativemotion of the particle compared to the time-averaged value.Ogawa (1978) introduced first the concept of granulartemperature Ts, where 3Ts is the mean square of particlevelocity fluctuations as expressed by equation (8), andSavage and Jeffrey (1981) made the first attempt to makemore substantial use of the ideas contained in the previoustheoretical work that had dealt with dense gases, forexample, Chapman and Cowling (1970).

Ts = 1

3〈u′2 + v′2 + w′2〉 (8)

In analogy with thermodynamic temperature, granulartemperature plays similar roles in generating pressuresand in governing the internal transport rates of mass,momentum, and energy.

Granular temperature can be generated with two dis-tinct mechanisms (Campbell, 1990). The first, the so-calledcollisional temperature generation, is a by-product of inter-particle collisions, in the sense that two colliding particles

DEBRIS FLOW 2177

will have resultant velocities depending not only on theirinitial velocities but also on the type of collision theyexperienced; therefore, they will contain apparently randomvelocity components. The second mode of temperature gen-eration, the so-called streaming temperature generation, isitself a by-product of the random particle velocities. Follow-ing its random path, a particle moving parallel to the localvelocity gradient will acquire an apparently random veloc-ity that is proportional to the difference in mean velocitybetween its present location and the point of its last colli-sion. It should be noted that in both mechanisms of granulartemperature generation, the magnitude of the generated ran-dom velocities is proportional to the local velocity gradient.However, unlike the collisional temperature generation, thestreaming mechanism can generate only the component ofrandom velocity lying in the direction perpendicular to themean velocity gradient, therefore the generated granulartemperature will be anisotropic.

Campbell (1990) describes how the physical similaritybetween rapid granular flows and kinetic-theory viewof gases has led to a great deal of work on creatingsimilar models for granular materials on the basis of theidea of deriving a set of continuum equations (typicallymass, momentum, and energy conservation) entirely frommicroscopic models of individual particle interactions. Allthe models are based on the assumption that particlesinteract by instantaneous collisions, implying that onlybinary or two-particle collisions need to be considered.Particles are usually modeled in a simple way, ignoringsurface friction or any other particle interactions tangentialto the contact-point, and considering a constant coefficientof restitution to represent the energy dissipated by theimpact normal to the point of contact between the particles,even though Lun and Savage (1986) and other researchersshowed a strong dependence of the coefficient of restitutionon the relative impact velocity. Furthermore, molecularchaos is generally assumed, this implying that the randomvelocities of particles are independently distributed.

Jenkins and Hanes (1998) apply kinetic theories to a sheetflow in which particles are driven by turbulent fluid andsupported by their collisional interactions rather than by thevelocity fluctuations of the interstitial fluid. Azanza et al.(1999) adapted kinetic theories to a channel flow in whichthe particles are driven by gravity. Armanini et al. (2005)accounted for the interstitial fluid interaction by means ofan added mass coefficient.

The constitutive relation for the particle pressures istherefore:

σ = Cρs

(1 + rρ

ρs

)(1 + 4Cg0)Ts where r = 1 + 2C

2(1 − C)(9)

The function g0(C) describes the variation of the particlecollision rate with concentration, and was derived byCarnahan and Starling (1969) from considerations about the

nearly geometric form of the virial series for nonattractingrigid spheres.

g0(C) = (2 − C)

2(1 − C)3 (10)

Again, the constitutive relation for the particle shearstress is taken to be the one for a dense molecular gas,in the form:

τ = −8D(1 + rρ/ρs)ρsC2g0Ts

1/2

5π1/2[1 + π

12

(1 + 5

8Cg0

)2]

du

dz(11)

The balance of particle fluctuation energy is equal to thatfor the energy of the velocity fluctuations of the moleculesof a dense gas. For inelastic particles, the gradient ofthe vertical component Q of the fluctuation energy flux,expressed by equation (13), is required to balance firstthe net rate of fluctuation energy production per unit ofvolume of the mixture, and secondly the rate of collisionaldissipation γd:

dQ

dz= τ

du

dz− γd (12)

where the first term (on the left-hand side) represents theenergy diffusion, the second one the net rate of production(the rate of working of the particle shear stress through themean shear rate) and the last one, γd, is the rate of colli-sional dissipation. Particles are driven into collisions by themean motion, creating fluctuation energy, while the inelas-ticity of the collisions dissipates fluctuation energy into realthermal energy.

The constitutive relation for the flux of particle fluctua-tion energy is taken to be the one for a dense moleculargas, in the form given by Chapman and Cowling (1970),while the rate of collisional dissipation per unit of volumemay be calculated using the Maxwellian velocity distribu-tion function, in the form given by Jenkins and Savage(1983):

Q = −4

[1 + 9π

32

(1 + 5π

12Cg0

)2]

C2g0

π0.5 ρs

(1 + rρ

ρs

)DTs

1/2 dTs

dz(13)

γd = 24(1 − e)C2g0

π0.5

ρs(1 + rρ/ρs)T3/2

s

D(14)

Savage (1998) developed a theory for slow, dense flowsof cohesionless granular materials for the case of pla-nar deformations, employing the notion of granular tem-perature. The conservation equations for mass, momen-tum, and particle fluctuation energy are employed. At

2178 OPEN-CHANNEL FLOW

(a) (c) (d)(b)

Increasing direction for flow slope and transport concentration

Increasing direction for Froude number

Channel slope

Froude number

Collisional

Frictional Frictional

FrictionalCollisional Frictional

Collisional

Figure 2 Typology of the flows examined: (a) loose bed, immature (or oversaturated) debris flow; (b) loose bed, maturedebris flow; (c) loose bed, plug (or undersaturated) debris flow; (d) rigid bed debris flow

low deformation rates, the apparent form of the con-stitutive behavior is similar to that of a liquid, in thesense that the actual viscosity decreases as the gran-ular temperature augments, contrary to rapid granularflows, in which viscosity increases as granular tempera-ture augments.

Stresses are constituted by two parts: a rate-independent,dry friction contribution, and a rate-dependent viscous part,having a quadratic dependence on the shear rate, obtainedfrom the high shear rate granular flow kinetic theories inthe form of equations (9) and (11). The magnitude of therate-independent contribution was chosen so that the sumof the two parts satisfied the overall momentum balanceperpendicular to the flow direction.

Savage and Jeffrey (1981) introduced a parameter R,involving the particle diameter D, the square root of thegranular temperature and the shear rate, written in theform:

R = D

T 1/2s

du

dz(15)

As discussed in Savage (1998), in both analyses andcomputer simulations, typically the parameter R is foundto be of the order of the unit for granular flows, rang-ing from purely collisional to slow, predominantly fric-tional flows.

Experimental Analysis

In contrast with previous experimental studies dealingwith solid–liquid flows in annular shear cells, closedducts, or nonrecirculatory chutes, the experimental anal-ysis carried out in a steady uniform channel flow hasshown that the dynamics of a free-surface debris flowconstituted of sedimentable material is more articulatedthan what was predicted by Bagnold for gravity-lessgranular flows. Contrary to what was presupposed byTakahashi (1978) in adapting Bagnold’s theory to debrisflows, in the experiments it has been observed thata debris flow can present a series of layers one ontop of the other, each governed by a different rheol-ogy. The experiments by Armanini et al. (2005) permit

to distinguish four main regimes of interest (Figure 2):(a) loose bed, immature debris flow; (b) loose bed, maturedebris flow; (c) loose bed, plug debris flow; and (d) rigidbed flow.

The first three regimes (a)–(c) exhibit loose-bed equilib-rium conditions. The immature debris-flow regime (a) ischaracterized by the flow of a clear water layer over afluid-driven sheet of granular material supported by con-tacts with the stagnant bed. For mature debris flow (b),the entire moving layer is composed of a mixture of liq-uid and grains. In the “plug” debris flow (c), a partiallyemerged, quasi-static assembly of grains translates over aliquid-granular shear layer.

The possibility for the flowing mixture to have an under-lying bed formed by the same constituents (movable bed )leads to the formation of an equilibrium condition, sincethe slope of the bed is a dependent variable dynami-cally coupled with the flow. For nonequilibrium condi-tions, the mixture has to flow over a nonerodible bed,and in uniform-flow conditions, the free surface is paral-lel to the rigid bed of the flume; therefore the presenceof the solid phase results somehow in a variable inde-pendent of the bed slope. In fact, in this case the soliddischarge is smaller than the effective transport capacityof the current. The distinction between equilibrium andnonequilibrium conditions has of course a strong influ-ence on the rheological processes throughout the crosssection of the uniform flow. In fact, considering the pro-files of the variables of interest in the stress formation (meanvelocity, velocity fluctuations, and solid concentration), onecan observe that they change enormously from cases withor without equilibrium. The experimental results made itpossible to create a picture of the flow rheology that israther simple: throughout the flow depth there are layersdominated by collisions among grains, whereas the comple-mentary domains are essentially frictional. When the tworheological mechanisms coexist, the separation betweencollisional and frictional layers is represented as a rathersharp interface, correspondent to Stokes numbers, definedby equation (16), being equal to 5 ÷ 10. Large Stokesnumbers characterize collisional regions, while small ones

DEBRIS FLOW 2179

characterize frictional layers.

St = 1

18

ρs

ρ

D2du/dz

ν(16)

In equation (16), ν is the kinematic viscosity of the inter-stitial fluid and D(du/dz) represents the relative velocitybetween adjacent sheared rows of particles.

Moreover, the experimental vertical profiles of energyproduction, diffusion, and dissipation (Armanini et al.,2003) lead to the conclusion that in the collisional lay-ers energy production balances roughly energy dissipa-tion, so that in equation (12) the diffusive term can beneglected. Exploiting this assumption, normal and shearstress described by equations (9) and (11) depend on thesquare of the shear rate, exactly like in Bagnold’s grain-inertia theory, even if the ratio between shear and normalstresses is perfectly constant only in Bagnold’s conjectures.

Global Relations for Debris Flows in Equilibrium

According to the descriptions above, a succession of fric-tional and collisional layers, one on top of the other,constitutes debris flows. However, considering the debrisflow throughout its depth, if the frictional layers, character-ized by small mean velocities, give a negligible contributionto the total discharge of the flow, uniform-flow formulascan be obtained referring only to the collisional/grain-inertia layers. For this purpose, it can be assumed (Taka-hashi, 1978) that the flow is steady and uniform and thatthe tangential projection of the entire burden of the flow(water + grains) is charged over the solid phase only, there-fore assuming that the shear stress in the liquid phaseis negligibly small. Moreover, the vertical component ofmomentum balance for the particle phase requires that thegradient in the particle pressure σ balances the buoyantweight of a unit volume of particles.

dσ(z)

dz= Cρ�g cos α

dτ

dz= (C� + 1)ρg sin α (17)

Under these hypotheses and considering the concentra-tion C to be constant (which corresponds to assume thedynamic friction angle ϕ to be constant), it is possible tointegrate Bagnold’s grain-inertia rheological equation (7)combined with equation (17) throughout the flow depth,thus obtaining a relation between the mean velocity and theslope of the current in uniform motion in the same formof Gaukler–Strickler or Chezy formula. In this case, Chezyfriction coefficient χDF is a function of the solid fraction,of the sediment size, of the ratio between fluid and soliddensity, of the friction angle ϕ of the material, and of the

flow depth h.

U = χDF

√h sin α with χDF = 2

5

h

λD

√g

ρ

ρs

(C� + 1)

a sin ϕ

(18)

Bagnold suggests to assume a = 0.042, while, on thebasis of debris-flow laboratory data, Takahashi (1978) pro-posed a greater coefficient a = 0.35. This value correspondsin case of real debris flows (e.g. for D = 5 cm, C = 0.56,ϕ = 36◦, � = 1.65, h = 3 m, C0 = 0.70, λ = 13) to aChezy coefficient χDF = 11 m1/2 s−1. The debris-flow con-centration, if assumed to be equal to that of incipientmovement in saturated conditions, can be expressed by thefollowing relation as a function of the channel slope and ofthe relative density of the material:

C = tan α

�(tan ϕ − tan α)(19)

Takahashi (1991) proposed empirical formulas similar toequation (18) when the regime is clearly not dominated bycollisional particle interactions.

In case of viscous debris flow or mudflows, the problemfluid is often treated as a single-phase visco-plastic fluid,characterized by a Herschel–Bulkley rheology (Coussot,1997). This kind of approach is particularly suitable whenthe presence of silt and clay is relevant. For low solidconcentrations, the flows behave like a Newtonian fluid,while by increasing the concentration the mixture presentsa yield stress τc and a nonlinear relation between the shearstress and the shear rate, as expressed by equation (20):

τ = τc + k

(du

dz

)n

if τ > τc

(du

dz

)n

= 0 if τ < τc (20)

where k and n are parameters depending on the type offluid and on its concentration, to be determined by meansof laboratory tests and by the utilization of a rheometer.Like equation (7), equation (20) expresses a rheologicallink between shear stress and shear rate, and therefore itcan be integrated throughout the flow depth in order toobtain the mean velocity value in uniform-flow conditions.

U = n

(n + 1)h

(ρmg sin α

k

)1/n (h − τc

ρmg sin α

)1+(1/n)

×[h − n

2n + 1

(h − τc

ρmg sin α

)](21)

where ρm represents the density of the homogeneousmixture. If we assume n = 1 and k = µ, relations (20) and(21) are referred to the case of a Bingham fluid.

2180 OPEN-CHANNEL FLOW

Debris-flow Peak Discharge

Takahashi (1978) proposed a scheme for the estimationof debris-flow peak discharge based on the hypothesisthat, after a short time, the debris-flow front assumes aself-similar profile. In this way, it is possible to makesome assessments regarding the ratio between liquid andsolid discharge.

The debris-flow front is supposed to move with velocityUf and also the tail goes downhill with uniform speed Ut,smaller than Uf; in order that debris-flow profile becomeslonger without becoming thicker.

With reference to Figure 3, the balances of liquid andsolid phase masses give:

U0h0 = Uf(1 − C)h − Ut(1 − C)h − Ut(1 − C0)sa + Uth0

(22)

UfCh = UtCh − UtC0a (23)

where h is the front depth, a is the depth of the excavationproduced by the passing of the tail, s is the degree ofsaturation of the pores under the debris flow. The liquidfeeding from upriver gives rise to a speed U0 and to a flowdepth h0 and then to a discharge per length unit q0 = U0h0.

From equation (23), we infer that the front speed must belesser than the tail speed (Ut < Uf), and then that the debrisflow itself tends to become longer while going downhill.

From equations (22) and (23), which admit that, as thedebris flow passes by, the material in the channel bedis completely saturated (s = 1), we derive the followingrelation between the debris-flow discharge and the incomingliquid discharge:

Ufh = U0hC0

C0 − C + h0

aC

(1 − U0

Uf

) (24)

Equation (24), relative to the propagation of a quasi-stationary debris-flow front, can be utilized to derive therelation between debris-flow discharge and incoming liquid

U f ∆t

a

(C0, s)

hC

U f

U t

U t ∆t

Q0

ah0t

t + ∆t

Figure 3 Scheme of quasi-uniform propagation ofdebris-flow front

discharge. Supposing that the front speed Uf coincides withthe speed of the incoming water, then relation (24) becomesthe following:

QDF = Q0C0

C0 − C(25)

where QDF is the debris-flow peak discharge and Q0 is theliquid peak discharge. As previously observed, in general,we can assume C0

∼= 0.65 for natural debris.According to Takahashi (1978), for sufficiently high

slopes (i > 20◦), debris-flow concentration can be assumedto be equal to 90% of the maximum concentration. By doingso, equation (25) becomes QDF = 10Q0.

In the case of milder slopes (i < 20◦), debris-flow con-centration is assumed to be equal to that of incipient move-ment in saturated conditions expressed by equation (19).

In practice, debris-flow concentration ranges between thefollowing limits:

0.3 <C

C0< 0.9 1.43Q0 < QDF < 10Q0 (26)

Liquid peak discharge Q0 can be calculated with com-mon hydrologic methods.

An alternative method used to determine debris-flowpeak discharge consists in assimilating the debris-flow frontto that which forms after the collapse of a natural dam. Theexpression obtained is a function of the torrent depth hm,to be evaluated upstream of the dam, and of the torrentwidth B.

QDF = 8

27Bhm

√ghm (27)

MATHEMATICAL MODELING

Mathematical models are tools suitable to describe thegeneral features of initiation, motion, and deposition ofa gravity-driven mixture of debris and water and, lastly,the zoning of areas where damage is likely to occur.Owing to the high complexity that characterizes a realevent, simplified assumptions in the mathematical modelinghave to be introduced. At present in the applications thegrain-size composition in the mixture is generally notaccounted for, and only the solid concentration, whichmay change in time and space, remains to characterize theflowing sediments. Therefore, the possibility to representthe variations of the sediment size throughout the debrisflow (from its steep front, mainly constituted of boulders,to the tail) is generally very limited.

General three-dimensional models require an extremelyhigh computing time and level of hardware to determinethe relevant solutions by means of numerical techniques,and are applicable, at present, to local situations only.

The representations in a reduced two- or one-dimensionalframe correspond to depth- or section-integrated models.

DEBRIS FLOW 2181

This averaging process requires hypotheses about the pro-files of the field variables (velocities, pressure, concentra-tion) through the flow depth. These assumptions make itpossible to derive models where the flow domain is thebasal topography.

The characterization of the fluid mixture takes intoaccount the fact that the fluid is not strictly homogeneous,but made up of different phases. The liquid phase refersto the interstitial slurry that is made up of water and ofthe finest grain fractions. The solid phase represents thecoarser grain fractions. Some models admit that the twophases have different velocity distributions. The couplingbetween the two phases is present both in the massconservations and in the momentum equations (due tothe solid–liquid stress interaction). When we apply thehypothesis that there is no relative velocity between thetwo phases, the resulting integrated models present onlyone momentum equation, referring to a fluid having thebulk mass density.

Since attention is focused on averaged equation models,the rheological properties of the flowing mixture is lumpedin the definition of algebraic expressions for the tan-gential stresses acting on the boundaries of the flowdomain, especially on the bottom. Therefore, the bound-ary stresses are a function of the field variables ofthe model (such as flow depth, velocity, slope, sedi-ment size) and show up as source terms in the momen-tum equations.

Mathematical and numerical models are thought of withproper boundary and initial conditions. Debris flows run-outand depositions over fans clearly depend on the incominghydrograph. Initial conditions may also play a significantrole, as in cases where the flow is triggered by an abruptcollapse of barrages (dam break–induced debris flows;Fraccarollo and Capart, 2002).

Motion equations are based on conservation laws. Theirnumber, along with the number of variables, depends on therefinement of the model. The options mainly concern thenumber of phases and the number of momentum equations,one for each direction and phase-velocity field. In case of anonhomogeneous fluid, the bed-level changes in time andthe morphological evolutions are part of the solution. Forsake of simplicity, one-dimensional models are referred to aflow section of unit width and rectangular shape, assumingno frictional influences from the sidewalls.

Modeling of a one-phase system in the one-dimensionalframework is herein reported:

∂∂t

(h) + ∂∂x

(Uh) = 0

∂∂t

(Uh) + ∂∂x

(βU 2h + g h2

2

)+ gh

∂zb∂x

= − τρm

(28)

where h is the flow depth, U is the flow-depth-averagedvelocity, ρm = ρ(C� + 1) is the density of the homoge-neous mixture, zb is bed elevation, τ /ρm represents the only

source term, providing the shear stress at the bottom, β isthe momentum-flux coefficient that takes into account thenonuniform velocity distribution through the flow depth.

In this context, the main point characterizing the fluid isthe formulation of the wall friction force, to be deducedfrom rheological assumptions. Many rheological modelshave been tested in the past by different authors. Mostof them can be classified in the following categories:(i) models for muddy fluid (e.g. Bingham models); (ii) mod-els for mixtures in the inertial regime (e.g. equation 11);(iii) a combination of different dissipation models, includ-ing strain-rate independent contributions, to simulate grain-to-grain static prolonged contact, and various strain-ratedependent ones, such as Newtonian, turbulent, or colli-sional behaviors.

Under the same hypotheses, and moreover assuming β =1, the two-dimensional extent of the mass and momentumbalances are:

∂∂t

(h) + ∂∂x

(hU) + ∂∂y

(hV ) = 0

∂∂t

(hU) + ∂∂x

(12gh2 + hU 2

)+ ∂

∂y(hUV ) + gh

∂zb∂x

= − τxρm

∂∂t

(hV ) + ∂∂x

(hUV ) + ∂∂y

(12gh2 + hV 2

)+ gh

∂zb∂y

= − τy

ρm(29)

where the inner shear stresses have been also neglected, asin most models of this kind, and where τx and τy are thecomponents of the bottom shear stress and U and V arethe components of the velocity.

One of the answers expected from model applicationsfor mapping the risk concerns the description of debrisdeposition phenomena at the alluvial fans. In case ofa homogeneous fluid, the stopping of the flow can beinduced by a visco-plastic constitutive law (i.e. Bingham,Herschel–Bulkley models): when the yield stress balancesor exceeds the local acting forces, the fluid comes to a localstop throughout the flow depth.

On the contrary, in situations where the debris depositioncomes together with a reduction of the water contentbecause of a velocity reduction of the flow over thealluvial fan, it is necessary to exploit models dealing witha nonhomogeneous two-phase fluid. The presence of solidand liquid phases is accounted for by considering thevolumetric solid concentration (depth averaged) in bothmass and momentum conservations. With a full two-phasemodel, two mass and two momentum equations shouldbe considered. However, assuming that the velocities ofthe two phases are correlated, it comes out that a singlemomentum equation (and a single velocity field), associatedto the bulk fluid-density, remains.

Here, it is chosen to represent the formally simpler case,assuming that there is no velocity-phase difference. The

2182 OPEN-CHANNEL FLOW

following model is obtained in the one-dimensional case:

∂∂t

(h + zb) + ∂∂x

(Uh) = 0∂∂t

(Ch + Cbzb) + ∂∂x

(CUh) = 0

∂∂t

[(C� + 1)Uh] + ∂∂x

[(C� + 1)

(βU 2h + kg h2

2

)]

+(C� + 1)gh∂zb∂x

= − τρ

(30)

where C is the depth-averaged solid concentration, ρ is thewater density, k is the active/passive coefficient employedin soil mechanics, determined by the depth-averaged ratiobetween the vertical and the longitudinal normal stresses(Savage and Hutter, 1991; Iverson, 1997); k is oftenassumed equal to one, as for pure fluids.

In two-phase models (equation 30) one more closure-assumption, relevant to the volumetric solid concentrationC, is needed with respect to the homogeneous fluid model(equation 28). This relation can be derived from assuminga transport-capacity formulation, under the hypothesis thatthe processes of debris transport rapidly fit the dynamicchanges (hypothesis referred to as equilibrium). Regard-ing the transport-capacity formulation to be used, equa-tion (19) is an example concerning granular debris flows.Many other empirical laws can be considered, spanningfrom refinements of bed-load formulations often employedin torrential floods (Armanini, 2005), to more specific

laws devoted to the massive sediment transport underconsideration.

Under the same hypotheses, assuming, moreover, that theβ and k coefficients are equal to unity, the two-dimensionalextension of the mass and momentum balances can bederived, in which, as in the case of homogeneous fluid,the inner shear stresses can be neglected.

MITIGATION AND RISK REDUCTIONMEASURES

The mitigation remedial and defence techniques againstdebris flows are in their present form relatively recenttools: in the past, in fact, people simply tried not tobuild in the areas where there had been previous debris-flow events.

Only recently, as a consequence of the frantic devel-opment of the tourism settlements, risk areas, frequentlylocated along the alluvial fans, have started to be urbanized.In general, the defence works against the debris flows canbe classified in two categories: the active countermeasuresand the passive countermeasures.

The former consist basically of interventions aiming atreducing the risk of debris-flow triggering. Then they aremeant to give stability to the debris deposits in the torrentsbeds or in the hillslopes. The latter instead are works builtto defend directly the settlements or the zones subject to

Consolidationof sediment deposits

Consolidation of bed torrent

Walls againstslope failures

Over passor tunnel

Channel

Deposition basin

Check dam anddebris flow breakers

Walls and embankments

Figure 4 Scheme of the different types of active defence works against debris flows

DEBRIS FLOW 2183

debris-flow risk or the single structures (Figure 4). Thereare of course works responding to both criteria.

Consolidation of the Debris Deposits

The stabilization of the debris deposits is obtained eitherby reducing the slope or by hindering or reducing the pos-sibility of saturation. This objective can be reached byconsolidation works transversal to the deposits (endowed,where possible, with effective draining systems), by pil-ing and anchors and, where possible, by vegetation.However, the bulkiest sources of the material mobi-lized by the debris-flow events are often located inplaces hardly reachable by the mechanical means; more-over, it is often counterproductive to move the depositsthemselves in order to build drainages, as the move-ment of the material may increase its instability. As aresult, the cases of applicability of these devices areoften limited.

Consolidation of Riverbed and Lateral Slopes

In case the debris deposit affects directly the channel bed,the more limited the extent of the works is, the easier isthe consolidation. The works constructed in these cases aresills or traditional closed check dams designed to reduceand to stabilize the riverbed slope and therefore to reducethe flow velocity. Often, in fact, the channel bed instabilityinitially gives rise to an intense bed load that can degenerateinto a debris flow owing to the erosion of the riverbanks orto the accumulations caused by the geometric variations ofthe sections (natural dam break). A system that augmentsthe channel bed stability is represented by the chains ofconsolidation dams. The check dams built for this purposeare absolutely similar to the check dams used in case ofsolid ordinary transport. A phenomenon that characterizesthe passing of the debris flow and that must be consideredin the dimensioning of the check dam foundations is theamplification of flow depth from bed fluidification thataccompanies the passing of the debris flow (Jaggi andPellandini, 1997).

Debris-flow Breakers

More often, open check dams are built to intercept the solidmaterial. However, debris flows usually occur in streamswith a steep slope, where there is often lack of spacewhere the material can be stored. In this case, the objec-tive of the check dam is to diminish the flow velocity inorder to reduce its destructive power in case of a dynamicimpact. Therefore, these structures must resist above allthe dynamic impact and they should also intercept thehuge boulders.

Various devices have been proposed for this purpose.The most common structures are slit check dams with

Figure 5 Slit check dam with a series of debris-flowbreaker on Torrent Chieppena – Trentino Italy (ProvinciaAutonoma di Trento, 2002). A color version of this imageis available at http://www.mrw.interscience.wiley.com/ehs

quite a large opening often protected by one or more stoutbuttresses that are dimensioned so that they can resist thedynamic impact (Figure 5).

The hydrodynamic working of such structures is notclear and practical criteria suggested by experience areused for their design. The most effective results areobtained by water separation because the reduction of waterconcentration increases considerably the energy dissipationinside the flow and consequently its velocity.

Yet it is important that the volume upstream of the checkdam remains free after each event, therefore the openingmust allow the removal of the material accumulated.Accordingly, it can be useful to dimension the opening,so that the excavators meant to remove the materialaccumulated can pass through it.

The upstream side of the buttresses is inclined withrespect to the vertical so as to reduce the dynamic impactof the debris flow that depends on the normal componentof velocity. The inclination adopted varies between 45%and 60%.

The overpressure �p generated in the dynamic impactof the debris-flow front colliding against a vertical wallis obtained by applying the mass balance and the motionquantity balance, thus obtaining (Armanini and Scotton,1993) the following expression:

�p = ρmU(U − aw) (31)

2184 OPEN-CHANNEL FLOW

where ρm is the debris-flow density, U is the mean velocityof the front and aw is the velocity of propagation of thegravitational wave reflected by the wall.

Very often relation (31) is corrected with an opportunecoefficient αp in order to consider possible secondaryeffects. The impact coefficient αp often includes the effectof the reflected wave:

�p = αpρsU2 (32)

The coefficient αp varies from 2 for quite slow and densedebris flows to 0.7 for faster and more liquid debris flows.

Debris Flows Artificial Channels and Retention

Basins

It has been already pointed out that often there is not suffi-cient space to intercept all the debris-flow volume upstreamurbanized areas. Provided that debris-flow breakers andcheck dams have reduced the flow velocity and interceptedpart of the largest boulders, it is necessary to canalize thetorrent downstream the retaining works, often by buildingartificial channels able to deviate the debris flow. Thesechannels usually cross the villages and the alluvial fanswhere the space available is limited. Yet, it is useful to dis-tinguish between a channel designed for the water dischargeonly and one affected by remarkable solid discharges or bydebris flows. Often, it is the second necessity that is moreurgent. Then the channel is supposed to resist to high veloc-ities and strong tangential stresses, and must offer a verystable bed. Therefore, it is often necessary to resort to chan-nels covered and strengthened with concrete. In this case,the covering surface is often smoothed to reduce the wallroughness. This is not the case of the channel for debrisflow (Figure 6) where the resistance does not depend onsurface roughness, but on particle collisions.

Figure 6 Artificial channel on Torrent Gola – Trentino Italy(Provincia Autonoma di Trento, 1991). A color version ofthis image is available at http://www.mrw.interscience.wiley.com/ehs

Bends with strong curvatures must be avoided in orderto prevent depositions induced by secondary effects due tothe supercritical nature of the flow, and in any case free-surface elevation must be properly accounted for, in orderto avoid lateral debris-flow flooding.

In other cases, it can be useful to deviate the debrisflow on the side to safeguard some sites. This deflectionis obtained by the same techniques used for the snowavalanches, that is, by diverting walls or dikes.

The topography of the torrent often does not offer spacesgranting a sufficient volume for the deposition of thesolid material. In this case, it is necessary to create suchstorages. The debris retention basins are created throughlateral training dikes, protected downstream by a checkdam, which, if necessary, is inserted into the body of anartificial banking.

To optimize the volume available, it is necessary toremember that the widening cone of an overcritical currentdepends on the Froude number of the incoming current.

Acknowledgment

The authors thank Prof. Riccardo Rigon for his help inpreparing the section about the triggering of debris flow.

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