Applicability of Kinematic, Diffusion, and Quasi-Steady Dynamic Wave Models to Shallow Mud Flows

Applicability of Kinematic, Diffusion, and Quasi-SteadyDynamic Wave Models to Shallow Mud Flows

Cristiana Di Cristo1; Michele Iervolino2; and Andrea Vacca3

Abstract: Unsteady shallow-layer flows may be described through full dynamic models or using simplified momentum equations, basedon kinematic, diffusion, and quasi-steady approximations, which guarantee a reduction of the computational effort. This paper aims toinvestigate through linear analysis the applicability range of simplified shallow-wave models with special concern to unsteady flows ofmud. Considering a three-equation depth-integrated Herschel-Bulkley model, the applicability of the approximated wave models is discussedcomparing the propagation characteristics of a small perturbation of an initial steady uniform flow as predicted by the simplified models withthose of the full dynamic model. Based on this comparison, applicability criteria for the different wave approximations for mud flows ofHerschel-Bulkley fluids, which account for the effects of the rheological parameters, are derived. The results show that accounting forthe fluid rheology is mandatory for the choice of an appropriate simplified model. DOI: 10.1061/(ASCE)HE.1943-5584.0000881.© 2014 American Society of Civil Engineers.

Author keywords: Herschel-Bulkley fluid; Unsteady flows; Shallow flow model; Simplified wave models.

Introduction

Flood waves may entrain huge quantities of sediment and evolveinto mud flows capable of catastrophic consequences, such as lossof lives and huge economic damages. Powerful mud flows can alsobe triggered by long or intense rains in mountain areas, resulting indebris avalanches, which inflict significant topographical changesand damages to the structures along banks and floodplains. There-fore, the prediction of debris or mud flows propagation represents akey aspect for managing and minimizing the related risks.

The characteristics of flows involving movement of solid par-ticles depend on many factors (such as suspended solid concen-tration, particle size and shape, pore pressure, etc.), and severaldifferent models have been proposed, based either on a two-phaseapproach (Iverson 1997; Pitman and Le 2005; Greco et al. 2012) oron a single-phase description of the flowing medium, considered asa homogeneous continuum. The latter category is adopted for mudwith high sediment concentration, which shows a very viscous andeven non-Newtonian behavior, along with the presence of a yieldstress (Coussot 1994). Among the different rheological models pro-posed, one of the most used is the viscoplastic one, characterizedby a yield stress above which a linear or nonlinear stress-strain re-lationship is assumed to hold. In the former case, the rheologicalmodel is referred to as linear viscoplastic or Bingham fluid, with

the yield stress and the viscosity increasing with solid concentration(Dent and Lang 1983; Liu and Mei 1989). On the other hand, thenonlinear viscoplastic (or Herschel-Bulkley fluid) model (Coussot1997) better reproduces the shear-thinning behavior of concen-trated mud, by means of a power-law stress-strain relationship withexponent smaller than unity (Huang and Garcia 1998).

Moreover, because in many geophysical situations variations inthe flow occur on a length scale much larger than flow thickness,the one-dimensional shallow-water approximation has been fruit-fully applied for both Bingham and Herschel-Bulkley models(Huang and Garcia 1997, 1998), similar to the classical Saint-Venant Equations (SVEs) for clear-water shallow flows.

The knowledge acquired in the field of clear-water flood routinghas demonstrated that SVEs (full dynamic wave model) may beefficiently substituted, under appropriate circumstances, by sim-plified equations known as kinematic, diffusion, and quasi-steadydynamic wave models. The applicability of these simpler floodrouting approaches has been investigated by estimating the magni-tude of the neglected terms in the dimensional and dimensionlessform of momentum equation (Fread 1983; Ferrick 1985; Moussaand Bocquillon 1996; Moramarco and Singh 2000), through thecomparison with numerical solutions of the dynamic model (Singh1994; Singh and Aravamuthan 1995; Moramarco and Singh 2002;Moramarco et al. 2008), or based on the analytical solution of thelinearized Saint-Venant Equations (LSVEs). In fact, the knowledgeof the analytical solution of LSVEs, which has been fruitfully usedto test numerical models (Venutelli 2011; Di Cristo et al. 2012a),to enlighten some peculiar flow features of the waves (Di Cristoand Vacca 2005; Ridolfi et al. 2006; Di Cristo et al. 2012b), to de-sign automatic controllers for open-channel systems (Litrico andFromion 2004), and to predict roll-waves occurrence (Montuori1963; Julien and Hartley 1986; Di Cristo et al. 2008, 2010), alsoallowed engineering criteria to be proposed for the applicationof these simplified models. Starting from the results of Ponce andSimons (1977) concerning the evolution of a perturbation on anuniform base flow in an unbounded channel, Ponce et al. (1978)proposed criteria for the applicability of the kinematic and diffu-sion models based on the perturbation dimensionless wave period.These criteria were derived comparing the wave celerity and the

1Assistant Professor, Dipartimento di Ingegneria Civile e Meccanica–Università di Cassino e del Lazio Meridionale, Via Di Biasio 43, 03043Cassino (FR), Italy (corresponding author). E-mail: [email protected]

2Assistant Professor, Dipartimento di Ingegneria Civile, Design,Edilizia ed Ambiente, Seconda Università di Napoli, Via Roma 29,81031 Aversa (CE), Italy. E-mail: [email protected]

3Associate Professor, Dipartimento di Ingegneria Civile, Design,Edilizia ed Ambiente, Seconda Università di Napoli, Via Roma 29,81031 Aversa (CE), Italy. E-mail: [email protected]

Note. This manuscript was submitted on January 20, 2013; approved onJuly 10, 2013; published online on July 12, 2013. Discussion period openuntil October 1, 2014; separate discussions must be submitted for indivi-dual papers. This paper is part of the Journal of Hydrologic Engineering,Vol. 19, No. 5, May 1, 2014. © ASCE, ISSN 1084-0699/2014/5-956-965/$25.00.

956 / JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

attenuation factor of the simplified wave models with those of thefull dynamic linearized approach.

Although different analyses based on the study of the linear-ized model in an unbounded channel have also been carried out(e.g., Menéndez and Norscini 1982; Dooge and Napiorkowski1987; Lamberti and Pilati 1996), the validity of the Ponce andSimons criteria has been further confirmed (Singh 1996). The gen-eralization of results of Ponce and Simons (1977) and Ponce et al.(1978) in the presence of the tail-water effect has been provided byTsai (2003).

Similarly to the clear-water case, simplified routing models mayrepresent an advantageous alternative to the numerical solution ofthe full dynamic equations for the prediction of mud flood propa-gation (e.g., O’Brien et al. 1993; Arattano and Savage 1994). How-ever, no criteria have been provided to identify the applicabilityrange of these simplified models, and this paper aims to partiallyfill this gap.

The study is carried out considering the depth-integrated, three-equation Herschel-Bulkley model (Huang and Garcia 1998), con-stituted by the continuity equation and two momentum equationsfor the plug and the shear layers, respectively, under laminar con-dition of flow. The two-layer schematization, different from thesingle-layer version (Coussot 1994; Ancey et al. 2012; Di Cristoet al. 2013a, c, d), may yield a more accurate description of the mudflow dynamics. Similar to the clear-water case (Ponce and Simons1977; Ponce et al. 1978), the present study considers an unboundeddomain and a sinusoidal perturbation of the initial uniform flow,and compares the propagation characteristics (wave celerity andattenuation factor) predicted by simplified models to those deducedby the full dynamic linearized model. The applicability conditionsof kinematic, diffusion, and quasi-steady dynamic wave approxi-mations, represented in simple maps, are presented for differentvalues of rheological parameters (i.e., the yield stress and thepower-law exponent).

Governing Equations

Consider a two-dimensional, unsteady, gradually varied, laminarmud flow down a plane of inclination θ with respect to the hori-zontal. In the following, the effect of the bed permeability is notaccounted for (Pascal 1999; Di Cristo et al. 2013b). If the character-istic flow depth is smaller than the characteristic flow length, theflow is governed by the boundary-layer approximation equations.Defining the x-axis and the z-axis along and normal to the planebed, respectively, the longitudinal velocity is denoted by u. Thehypothesis of hydrostatic pressure distribution along the z-axis isalso assumed. Adopting the Herschel-Bulkey model (Huang andGarcia 1998), the following relations are considered:

μn

���� ∂u∂z����nsgn

�∂u∂z

�¼ τ s − τ ysgn

�∂u∂z

�if τ s ≥ τ y ð1Þ

μn

���� ∂u∂z����nsgn

�∂u∂z

�¼ 0 if τ s < τ y ð2Þ

where τ s = shear stress; τ y = yield stress; μn = dynamic viscosity;and n = flow index, ranging between 0 and 1 for a shear-thinningfluid. In uniform flow, the shear stress, assumed zero at the freesurface, increases linearly to its maximum at the bottom. If theshear stress at the bed τb is larger than the yield stress, a yield sur-face where the local shear stress equals the yields stress (τ s ¼ τ y)divides the flow depth h into two layers: a shear layer with thick-ness hs where τ s > τ y and a plug layer where τ s ≤ τ y. In the plug

layer (hs ≤ z ≤ h), the strain rate is zero and the velocity is con-stant, while in the shear layer (0 ≤ z ≤ hs), the following velocitydistribution is assumed:

u ¼ Up

�1 −

�1 − z

hs

�ðnþ1Þ=n�ð3Þ

with Up the velocity in the plug layer, expressed as

Up ¼ nnþ 1

�ρghnþ1

s sin θμn

�1=n

ð4Þ

where g = gravity acceleration; and ρ = fluid density. The flow rateover the entire depth, q, is expressed as

q ¼Z

h

0

udz ¼ Up

�h − n

2nþ 1hs

�ð5Þ

Applying the von Karman’s momentum integral method, alongwith the kinematic boundary conditions (Huang and Garcia 1998),the following depth-integrated equations are obtained:

∂h∂t þ

∂q∂x ¼ 0 ð6Þ

∂Up

∂tðaÞ

þUp∂Up

∂xðbÞ

þ g

�∂h∂xðcÞ

cos θ − sin θðdÞ

�þ τ yρðh − hsÞ

ðeÞ

¼ 0 ð7Þ

α1hs∂Up

∂t þ ðα1 − 1ÞUp∂hs∂t|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðaÞ

þ ðα2 − α1ÞU2p∂hs∂x þ ð2α2 − α1ÞhsUp

∂Up

∂x|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ðbÞ

þ ghs

�∂h∂x cos θ

ðcÞ− sin θ

ðdÞ

�þ μn

ρ

�nþ 1

n

Up

hs

�n

ðeÞ

¼ 0 ð8Þ

with t = time; and α1 ¼ nþ 1=

�2nþ 1

�and α2 ¼

2ðnþ 1Þ2=ð2nþ 1Þð3nþ 2Þ representing two shape factors. Theterms in Eqs. (7) and (8) represent the contributions of (a) localinertia, (b) convective inertia, (c) pressure gradient, (d) gravity, and(e) friction. For a uniform flow satisfying the condition τb > τ y, theyield depth may be defined as hy ¼ τ y=ðρg sin θÞ, which representsthe minimum depth for a uniform layer of mud to start flowing ona slope.

The system of Eqs. (6)–(8) is normalized using the followingdimensionless variables (Huang and Garcia 1998):

~h ¼ hho

; ~hs ¼hsho

; ~x ¼ xlo; ~t ¼ uo

lot;

~Up ¼ Up

uo; F ¼ uoffiffiffiffiffiffiffi

ghop ; α ¼ hy

hoð9Þ

with lo ¼ ho cot θ and

uo ¼n

2nþ 1

�ρghnþ1

o sin θμn

�1=n

ð10Þ

which represents the depth-averaged velocity for a steady, uniformflow of a power-law fluid with depth ho. The dimensionless param-eter α, which represents the ratio of the yield stress to the bottom

JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014 / 957

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

stress of a uniform flow of depth ho, has to be less than unity toguarantee the existence of a shear layer.

Omitting the superscript, in the limit of small bed slope(cos θ ≈ 1), the dimensionless form of Eqs. (6)–(8) is

∂h∂t þ

∂∂x ½Upðhþ ðα1 − 1ÞhsÞ� ¼ 0 ð11Þ

F2

�at∂Up

∂t þ acUp∂Up

∂x�þ ap

∂h∂x þ af

�α

ðh − hsÞ− 1

�¼ 0

ð12Þ

F2

�at

�α1hs

∂Up

∂t þ ðα1 − 1ÞUp∂hs∂t

�

þ ac

�ðα2 − α1ÞU2

p∂hs∂x þ ð2α2 − α1ÞhsUp

∂Up

∂x��

þ aphs∂h∂x þ af

�αn1

�Up

hs

�n − hs

�¼ 0 ð13Þ

The indices at, ac, ap, af , integer coefficients of values zero orone, are introduced to select the different approximations of the fulldynamic model: at allows to switch off the contribution of localinertia, ac that of convective inertia, ap that of pressure gradient,and af that of friction.

The system (11)–(13), in its complete form, represents thegoverning equations for shallow, laminar mud flows following theHerschel-Bulkley model, with the Bingham model as a particularcase for n ¼ 1 (Liu and Mei 1989). Depending on the terms re-tained in Eqs. (12) and (13), the following well-known approxima-tions are obtained:• Kinematic wave model (KWM) for at ¼ ac ¼ ap ¼ 0 and

af ¼ 1;• Diffusion wave model (DWM) for at ¼ ac ¼ 0 and

ap ¼ af ¼ 1; and• Quasi-steady dynamic wave model (QSWM) for at ¼ 0 and

ac ¼ ap ¼ af ¼ 1

Linearized Full Dynamic Model

Consider a steady uniform flow characterized by

ho ¼ 1; hso ¼ 1 − α

Upo ¼1

α1

ð1 − αÞα3 with α3 ¼nþ 1

nð14Þ

An infinitesimal disturbance is then introduced, with a normalmode ðh 0; h 0

s;U 0pÞ ¼ ðh; hs; UpÞeiðkx−ωtÞ, where k ¼ 2π=λ is the

real dimensionless wave number, with λ the dimensionless wave-length and ω the complex dimensionless propagation number. Fromthe linearized perturbed equations, the following dispersion relationis obtained:

η1ω3 þ ðη2 þ Iη3Þω2 þ ðη4 þ Iη5Þωþ η6 þ Iη7 ¼ 0 ð15Þwith

η1 ¼ a2t ηF4ð1 − α1Þ ð16aÞ

η2 ¼ atη2F4k½acðα2 − 1Þ þ atðα1 − 1Þ� ð16bÞ

η3 ¼atafF2½α1ð1 − αÞ þ αðnþ 1Þ�

αð16cÞ

η4 ¼ atap ηk2F2ðα1 − 1Þ

þ fη3F4k2ac½atð1 − α2Þ þ acðα1 − α2Þ�g −a2fnð1 − αÞ

αη

ð16dÞ

η5 ¼−afηF2k

�at1−α1

αþ1−α

α½acð2α2−α1Þþatð2α1−1Þ�

þðnþ1ÞðacþatÞþap acη2k3F2½ð1−αÞð1−α1Þð1−α2Þ

þðα1−α2Þ� ð16eÞ

η6 ¼ a2fk

�α1ð1−αÞð2nþ1Þ− ð1−αÞðnþ1Þþa2fðnþ1Þ

α

�

þa2cη4k3F4ðα2−α1Þþapacη2k3F2½ð1−αÞð1−α1Þð1−α2Þþðα1−α2Þ� ð16fÞ

η7 ¼ k2af

�acη2F2

α½ð1 − αÞðα2 − α1 þ α1α2Þ þ αðnþ 1Þ

þ ðα1 − α2Þ� þapα½αð1 − αÞð1 − α1Þð2nþ 1Þ

þ −αðnþ 1Þ þ ð1 − αÞ2ð1 − α1Þ − ð1 − αÞ�

ð16gÞ

in which η ¼ ð1 − αÞα3=α1.The third-degree complex characteristic [Eq. (15)] in ω de-

scribes the propagation of an infinitesimal disturbance superposedto a uniform Herschel-Bulkley mud flow, with a dimensionlesswave-phase speed or celerity c ¼ RðωÞ=k ¼ λ=τ, with τ thedimensionless wave period, and σ ¼ IðωÞ the growth rate. For anygiven triplet (F, α, n) and k value, Eq. (15) furnishes three roots forω and correspondingly three celerities, ci, and three growth rates,σiðwith i ¼ 1; 2; 3Þ. In what follows, attention is focused on theprimary wave, which propagates downstream with the highestcelerity (c1 > 0), in a stable flow condition (σ1 < 0 or F < Fc, withFc the limiting stability Froude number).

Fig. 1 reports the decay rate, −σ1 [Fig. 1(a)] and the dimension-less celerity, c1 [Fig. 1(b)] of the primary dynamic wave as a func-tion of the dimensionless wave period, τ , for α ¼ 0.5 and n ¼ 0.4,and for three different Froude values (F ¼ 0.21, F ¼ 0.58, F ¼1.0). The limiting stability Froude value for the considered (α; n)pair is Fc ¼ 1.4. The investigated wave-number range is 0 < k ≤1,000, while results are represented in graphical form in the range0.01 ≤ τ ≤ 1,000.

Independently of the F value, both quantities −σ1 and c1 aredecreasing functions of the dimensionless wave period. In particu-lar, as τ increases, the decay rate approaches zero, while the celerityreaches an asymptotic (long-wave) value:

clw ¼ ðnþ 1Þηnð1 − αÞ ¼

ð2nþ 1Þð1 − αÞ1=nn

ð17Þ

Expression (17) shows that the long-wave celerity depends onthe rheological parameters (n;α), whereas it does not depend on F.Conversely, for τ ≤ 10, the Froude number has a strong influenceof both −σ1 and c1 (i.e., a decrease in F increases both the decayrate and the celerity of the wave).

958 / JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Simplified Models versus Full Dynamic

With reference to the primary wave characteristics, based on pre-vious analyses of clear-water flows (Ponce et al. 1978; Tsai 2003),the applicability of the simplified models is evaluated by meansof the celerities ratio CSM ¼ cFDM=cSM and the attenuation factorsratio DSM ¼ δ �FDM =δ�SM, where δ� ¼ eδ with

δ ¼ 2πIðωÞRðωÞ ¼ 2π

σkc

ð18Þ

with subscripts FDM and SM that refer to the full dynamic modeland to a generic simplified model, respectively. In the following, fora fixed fluid the CSM andDSM ratios are represented as a function ofthe dimensionless wave period τ for different F values. Moreover,the applicability domain of each simplified model has been definedas the region of the (F=Fc; τ ) plane in which the estimates of SMdeviate from FDM ones less than 5% (i.e., jDSM − 1j ≤ 5% orjCSM − 1j ≤ 5% (Ponce et al. 1978; Tsai 2003).

Kinematic Wave Model

Considering the kinematic wave model (KWM), Eq. (15) reducesto the real first-degree equation:

η4ωþ η6 ¼ 0 ð19Þwith

η4 ¼ − nð1 − αÞαη

ð20aÞ

η6 ¼ k

�α1ð1 − αÞð2nþ 1Þ þ αðnþ 1Þ

α

�ð20bÞ

Similar to the clear-water case, the kinematic model is char-acterized by the propagation of a single wave with neither am-plification nor decay. The KWM celerity cKWM ¼ −η6=ðkη4Þcorresponds to the asymptotic value clw of the FDM [Eq. (18)];therefore, it does not depend on F and k, but only on the rheologicalparameters. To focus on the main feature of KWM, Fig. 2 repre-sents the CKWM ratio versus the dimensionless wave period, for thesame F values and rheological parameters of Fig. 1. As expected,

for high values of τ , the ratio tends to 1, while for small values of τ ,the KWM always underestimates the celerity. Such an underesti-mation becomes more evident as F decreases. As far as the waveattenuation is concerned, because σKWM ¼ 0, KWM represents avalid approximation of the FDM only for high values of τ , whenthe decay rate of the primary wave is close to 0 [Fig. 1(a)].

The minimum value of the dimensionless wave period abovewhich the KWM becomes applicable, in terms of celerity (τCKWM)and attenuation factor (τDKWM), strongly depends on the rheologicalparameters, along with the Froude number value. Fig. 3(a) reportsthe τCKWM as a function of the ratio F=Fc for α ¼ 0.5 and differentn values, and Fig. 3(b) refers to n ¼ 0.4 and different α values.For a fixed ðα; nÞ pair, each curve represents the lower boundaryof the KWM celerity applicability domain. Similarly, Fig. 4 showsthe equivalent set of curves for τDKWM. In Table 1, the limiting sta-bility Froude values, Fc, for the considered (α; n) pairs are given.For the sake of comparison, in Figs. 3 and 4, the curves based onthe analysis by Ponce and Simons (1977) for turbulent clear-waterflows are also reported, showing that the applicability domain of

Fig. 1. (a) Primary wave dimensionless decay rate; (b) celerity as a function of the dimensionless wave period for different F values (α ¼ 0.5and n ¼ 0.4)

Fig. 2. Ratio of full dynamic to kinematic wave celerity (CKWM) asa function of the dimensionless wave period for different F values(α ¼ 0.5 and n ¼ 0.4)

JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014 / 959

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

the KWM for mud flows may strongly differ from that of the cor-responding clear-water domain.

For high values of the Froude number (i.e., for F → Fc),because the rate of decay predicted by the FDM vanishes, theKWM applicability domain based on the prediction of attenuationfactor enlarges (Fig. 4), independently of the rheological parame-ters. In contrast, for small values of the Froude number, τDKWM in-creases because the maximum value of the ratio DKWM is expected

for F → 0. Figs. 3 and 4 illustrate that as far as both celerity andthe attenuation factor are concerned, the applicability domain ofthe KWM enlarges with n and contracts with α, and that the con-dition on the attenuation factor is the most restrictive. For a givenfluid, the limiting values ~τCKWM ¼ max0≤F≤Fc

τCKWM and ~τDKWM ¼max0≤F≤Fc

τDKWM represent a conservative estimate of the lowerbounds for the applicability of KWM. Figs. 5(a and b), which re-port the ~τCKWM and ~τDKWM values for several rheological parametervalues, roughly define for each fluid the minimum τ value abovewhich the kinematic approximation can be safely used. For the sakeof comparison, the reference values of ~τCKWM and ~τDKWM for clearwater are also represented, showing that mud flow rheology iscrucial for assessing the applicability of KWM, especially if highvalues of α and small values of n are expected.

Diffusion Wave Model

For the diffusion wave model (DWM), Eq. (15) reduces to thecomplex first-degree equation:

η4ωþ η6 þ Iη7 ¼ 0 ð21Þ

Fig. 3. Applicability of the KWM in terms of celerity: (a) α ¼ 0.5; (b) n ¼ 0.4

Fig. 4. Applicability of the KWM in terms of attenuation factor: (a) α ¼ 0.5; (b) n ¼ 0.4

Table 1. Limiting Stability Froude Number Values for DifferentRheological Parameters

α n Fc

0.5 0.2 4.900.5 0.4 1.400.5 0.8 0.890.5 1.0 0.840.1 0.4 0.210.2 0.4 0.410.4 0.4 0.920.6 0.4 2.40

960 / JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

with

η7 ¼ k2�1

α½αð1 − αÞð1 − α1Þð2nþ 1Þ − αðnþ 1Þ

þ ð1 − αÞ2ð1 − α1Þ − ð1 − αÞ�

ð22Þ

and the expressions of η4 and η6 equivalent to those of the KWM[Eq. (20)]. Therefore, the DWM again predicts a single wave thatpropagates with celerity cDWM ¼ cKWM but with a decay rate−σDWM ¼ η7=η4. Fig. 6, which shows the attenuation factor ratioDDWM as a function of τ for α ¼ 0.5, n ¼ 0.4, and F ¼ 0.21, 0.58,1.0, indicates that the DWM is able to reproduce the FDM resultsonly for long waves.

Because cDWM ¼ cKWM, the applicability domain for thecelerity prediction of the diffusive model may be deduced fromFigs. 4(a) and 5(a). In contrast, the threshold value τDDWM, abovewhich the DWM predicts the attenuation factor within the pre-scribed accuracy of 95%, is illustrated in Fig. 7. Fig. 7(a), wherea constant α value has been considered, shows that similarly tothe KWM, the domain of applicability of DWM enlarges as n

Fig. 5. Applicability map for the KWM in terms of (a) celerity; (b) attenuation factor

Fig. 6. Ratio of full dynamic to diffusion wave attenuation factors(DDWM) as a function of the dimensionless wave period for differentF values (α ¼ 0.5 and n ¼ 0.4)

Fig. 7. Applicability of the DWM in terms of attenuation factor: (a) α ¼ 0.5; (b) n ¼ 0.4

JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014 / 961

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

increases, while for a fixed value of n, the yield stress has anopposite effect [Fig. 7(b)]. The curves corresponding to turbulentclear water are also represented to demonstrate the effect of thedifferent rheologies on the applicability domain of the DWM formud flows.

Fig. 8 reports the maximum ~τDDWM values for several (α; n) pairs,along with the corresponding clear-water reference value. It canbe deduced that the applicability τ -range of the DWM for mudflows is, independent of the n values, much smaller than the cor-responding one for clear water except for small values of α. Finally,the comparison between Figs. 8 and 5(b) suggests that in termsof attenuation factor, the range of applicability of DWM is widerthan that of KWM, because the ~τDDWM values are smaller thanthose of ~τDKWM.

Quasi-Steady Dynamic Wave Model

Using the quasi-steady dynamic wave model (QSWM), Eq. (15)reduces to the following complex first-degree equation:

ðη4 þ Iη5Þωþ η6 þ Iη7 ¼ 0 ð23Þ

with

η4 ¼ η3β2k2ðα1 − α2Þ − nð1 − αÞαη

ð24aÞ

η5 ¼ −ηF2k

�1 − αα

ð2α2 − α1Þ þ nþ 1

�ð24bÞ

η6 ¼ k

�α1ð1 − αÞð2nþ 1Þ

αþ nþ 1

�þ η4k3F4ðα2 − α1Þ

þ η2k3F2½ð1 − αÞð1 − α1Þð1 − α2Þ þ ðα1 − α2Þ� ð24cÞ

η7 ¼ k2�η2F2

α½ð1 − αÞðα2 − α1 þ α1α2Þ þ αðnþ 1Þ

þ ðα1 − α2Þ� þ1

α½αð1 − αÞð1 − α1Þð2nþ 1Þ

− αðnþ 1Þ þ ð1 − αÞ2ð1 − α1Þ − ð1 − αÞ�

ð24dÞ

As for the previous simplified models, the QSWM predicts theexistence of only one wave that is characterized by the followingcelerity and decay rate:

cQSWM ¼ − η6η4 þ η7η5kðη24 þ η25Þ

ð25Þ

−σQSWM ¼ η4η7 − η6η5η24 þ η25

ð26Þ

Eqs. (25) and (26) show that both the celerity, cQSWM, and thedecay rate, −σQSWM are functions of k and depend on both F andthe rheological parameters. Fig. 9 depicts the ratios and as a func-tion of τ , again for α ¼ 0.5 and n ¼ 0.4, and for the same threeFroude number values considered previously. Similar to the DWM,the QSWM leads to accurate predictions for long waves, in termsof both celerity and decay rate, whereas for low τ, it largely under-estimates propagation characteristics.

The lower bounds of the wave period above which the ap-plication of the QSWM leads to accurate predictions, for the

Fig. 9. Ratio of full dynamic to quasi-steady: (a) dynamic wave celerity; (b) attenuation factor as a function of the dimensionless wave period fordifferent F values (α ¼ 0.5 and n ¼ 0.4)

Fig. 8. Applicability map for the DWM in terms of attenuation factor

962 / JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

celerity (τCQSWM) and the attenuation factor (τDQSWM), are reported as

functions of F=Fc in Figs. 10 and 11, respectively. In Figs. 10(a)and 11(a), the effect of the parameter n is demonstrated, whereasFigs. 10(b) and 11(b) illustrate the dependence on the dimension-less yield stress. From both qualitative and quantitative points ofview, the results provided by the QSWM do not differ substantiallyfrom those pertaining to the KWM for the celerity and to the DWMfor the attenuation factor. As previously noted, the applicabilitydomain expands with n and reduces with α for the QSWM.

Figs. 12(a and b) report the values of the maxima ~τCQSWM

and ~τDQSWM for several (n;α) pairs and defines the range of appli-cability of the QSWM for a mud flow. The comparison betweenFigs. 12(a and b) indicates that for the QSWM, ~τCQSWM < ~τDQSWM;hence, the range of applicability of QSWM based on the celerityprediction is larger than that for attenuation rate. Comparison withthe clear-water results, deduced from the analysis by Ponce andSimons (1977), shows that, differently from the kinematic anddiffusive wave models, the QSWM criteria for clear water areconservative for mud flows characterized by small values of α andhigh values of n

Final Remarks and Models Comparison

Following the criterion proposed for clear water by Ponce et al.(1978), approximate wave models for mud flows may be confi-dently applied to predict wave celerity and attenuation factor, pro-vided that the dimensionless flood rising time is larger than thethreshold values of the dimensionless wave period ~τCSM and ~τDSM.As far as kinematic, diffusive, and quasi-steady simplified modelsare concerned, these thresholds are given for different mud proper-ties in Figs. 5, 8, and 12, respectively.

To illustrate the practical impact of the present analysis, thedirect comparisons of the applicability minimum threshold values~τCSM and ~τDSM of the three models are provided in Table 2 for severalfluids. Table 2 indicates that independent of the rheological param-eters, for all simplified models the requirement for predicting theattenuation factor is more restrictive than that necessary for thecelerity. Moreover, because the DWM is characterized by the low-est threshold dimensionless wave periods for both celerity andattenuation factor, this simplified model may deal with floodswith smaller rising time, and it therefore appears as the mostsuitable one.

Fig. 10. Applicability of the QSWM in terms of celerity: (a) α ¼ 0.5; (b) n ¼ 0.4

Fig. 11. Applicability of the QSWM in terms of attenuation factor: (a) α ¼ 0.5; (b) n ¼ 0.4

JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014 / 963

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Conclusions

The applicability of the kinematic, diffusion, and quasi-steady dy-namic wave models to unsteady mud flows has been investigatedthrough linear analysis, by comparing the propagation characteris-tics of the simplified models to those of the full dynamic version,represented by a three-equation depth-integrated Herschel-Bulkleymodel. The presented criteria, expressed as minimum value of thedimensionless wave period, τ , identify the conditions for whichsimplified wave models are able to describe the wave-propagationcharacteristics in mud flows with accuracy above 95%. Notewor-thy, the requirement for the applicability of the simplified model isthat the dimensionless rising time of the flood hydrograph has tobe larger than the dimensionless wave period threshold values forthe celerity ~τCSM and the attenuation factor ~τDSM. Moreover, theanalysis suggests that the criteria developed for water flows inturbulent condition cannot be straightforwardly applied to mudflows, because fluid rheology influences the propagation character-istics and significantly affects the minimum value of the dimension-less wave period required for the applicability of the differentapproximations. The criteria provided in this study therefore re-present a guideline to verifying if, for a particular mud flow, asimplified model provides a sufficiently accurate approximationof the propagation characteristics predicted by the full dynamicmodel. The reported simple maps may be useful for engineeringpredictions.

References

Ancey, C., Andreini, N., and Epely-Chauvin, G. (2012). “Viscoplasticdambreak waves: Review of simple computational approaches andcomparison with experiments.” Adv. Water Resour., 48, 79–91.

Arattano, M., and Savage, W. Z. (1994). “Modelling debris flow askinematic waves.” Bull. Int. Assoc. Eng. Geol., 49(1), 3–13.

Coussot, P. (1994). “Steady, laminar, flow of concentrated mud suspensionsin open channel.” J. Hydraul. Res., 32(4), 535–559.

Coussot, P. (1997). Mudflow rheology and dynamics. Balkema, Leiden,The Netherlands.

Dent, J. D., and Lang, T. E. (1983). “A biviscous modified Bingham modelof snow avalanche motion.” Ann. Glaciol., 4, 42–46.

Di Cristo, C., Iervolino, M., and Vacca, A. (2012a). “Discussion of ‘Analy-sis of dynamic wave model for unsteady flow in an open channel.’”J. Hydraul. Eng., 10.1061/(ASCE)HY.1943-7900.0000538, 915–917.

Di Cristo, C., Iervolino, M., and Vacca, A. (2012b). “Green’s function ofthe linearized Saint–Venant equations in laminar and turbulent flows.”Acta Geophys., 60(1), 173–190.

Di Cristo, C., Iervolino, M., and Vacca, A. (2013a). “Boundary conditionseffect on linearized mud-flow shallow model.” Acta Geophys., 61(3),649–667.

Di Cristo, C., Iervolino, M., and Vacca, A. (2013b). “Gravity-driven flow ofa shear-thinning power-law fluid over a permeable plane.” Appl. Math.Sci., 7(33), 1623–1641.

Di Cristo, C., Iervolino, M., and Vacca, A. (2013c). “On the applicability ofminimum channel length criterion for roll-waves in mud-flows.” J. Hy-drol. Hydromech., 61(4), 286–292.

Di Cristo, C., Iervolino, M., and Vacca, A. (2013d). “Waves dynamicsin a linearized mud-flow shallow model.” Appl. Math. Sci., 7(8),377–393.

Di Cristo, C., Iervolino, M., Vacca, A., and Zanuttigh, B. (2008).“Minimum channel length for roll-wave generation.” J. Hydraul. Res.,46(1), 73–79.

Di Cristo, C., Iervolino, M., Vacca, A., and Zanuttigh, B. (2010). “Influenceof relative roughness and Reynolds number on the roll-waves spatialevolution.” J. Hydraul. Eng., 10.1061/(ASCE)HY.1943-7900.0000139,24–33.

Di Cristo, C., and Vacca, A. (2005). “On the convective nature of roll wavesinstability.” J. Appl. Math., 2005(3), 259–271.

Dooge, J. C. I., and Napiorkowski, J. J. (1987). “Applicability of diffusionanalogy in flood routing.” Acta Geophys. Pol., 35(1), 65–75.

Ferrick, M. G. (1985). “Analysis of river wave types.” Water Resour. Res.,21(2), 209–220.

Fread, D. L. (1983). “Applicability criteria for kinematic and diffusion rout-ing models.” HRL 183, Hydrologic Research Laboratory, NationalWeather Service, National Oceanic and Atmospheric Administration,Silver Spring, MD.

Greco, M., Iervolino, M., Leopardi, A., and Vacca, A. (2012). “A two-phasemodel for fast geomorphic shallow flows.” Int. J. Sediment Res., 27(4),409–425.

Fig. 12. Applicability map for the QSWM in terms of (a) celerity; (b) attenuation factor

Table 2. Minimum Values of the Dimensionless Wave Period for theSimplified Model Applicability for Different n Values (α ¼ 0.5)

n ~τCKWM ¼ ~τCDWM ~τCQSWM ~τDKWM ~τDDWM ~τDQSWM

0.2 74 77 3,146 2,871 3,2000.4 18 20 761 707 8100.8 9 11 375 342 4461.0 8 10 318 298 392

964 / JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Huang, X., and Garcia, M. H. (1997). “A perturbation solution forBingham-plastic mudflows.” J. Hydraul. Eng., 10.1061/(ASCE)0733-9429(1997)123:11(986), 986–994.

Huang, X., and Garcia, M. H. (1998). “A Herschel-Bulkley model for mudflow down a slope.” J. Fluid Mech., 374, 305–333.

Iverson, R. M. (1997). “The physics of debris flows.” Rev. Geophys., 35(3),245–296.

Julien, P. Y., and Hartley, D. M. (1986). “Formation of roll waves in laminarsheet flow.” J. Hydraul. Res., 24(1), 5–17.

Lamberti, P., and Pilati, S. (1996). “Flood propagation models for real-timeforecasting.” J. Hydrol., 175(1–4), 239–265.

Litrico, X., and Fromion, V. (2004). “Simplified modeling of irrigationcanals for controller design.” J. Irrig. Drain. Eng., 10.1061/(ASCE)0733-9437(2004)130:5(373), 373–383.

Liu, K. F., and Mei, C. C. (1989). “Slow spreading of a sheet of Binghamfluid on an inclined plane.” J. Fluid Mech., 207, 505–529.

Menéndez, A. N., and Norscini, R. (1982). “Spectrum of shallow waterwaves.” J. Hydr. Div., 108(1), 75–94.

Montuori, C. (1963). “Discussion of ‘Stability aspects of flow in openchannels.’” J. Hydr. Div., 89(HY4), 264–273.

Moramarco, T., Pandolfo, C., and Singh, V. P. (2008). “Accuracy ofkinematic wave and diffusion wave approximations for flood routing.I: Steady analysis.” J. Hydrol. Eng., 10.1061/(ASCE)1084-0699(2008)13:11(1078), 1078–1088.

Moramarco, T., and Singh, V. P. (2000). “A practical method for analysisor river waves and for kinematic wave routing in natural channelnetworks.” Hydrol. Process., 14(1), 51–62.

Moramarco, T., and Singh, V. P. (2002). “Accuracy of kinematic wave anddiffusion wave for spatial-varying rainfall excess over a plane.” Hydrol.Process., 16(17), 3419–3435.

Moussa, R., and Bocquillon, C. (1996). “Criteria for the choice of floodrouting methods in natural channels.” J. Hydrol., 186(1–4), 1–30.

O’Brien, J. S., Julien, P. Y., and Fullerton, W. T. (1993). “Two-dimensionalwater flood and mudflow simulation.” J. Hydrol. Eng., 10.1061/(ASCE)0733-9429(1993)119:2(244), 244–261.

Pascal, J. P. (1999). “Linear stability of fluid flow down a porous inclinedplane.” J. Phys. D, 32(4), 417–422.

Pitman, E. B., and Le, L. (2005). “A two-fluid model for avalanche anddebris flows.” Phil. Trans. R. Soc. A, 363(1832), 1573–1601.

Ponce, V. M., Li, R. M., and Simons, D. B. (1978). “Applicability ofkinematic and diffusion models.” J. Hydr. Div., 104(HY3), 353–360.

Ponce, V. M., and Simons, D. B. (1977). “Shallow water propagation inopen channel flow.” J. Hydr. Div., 103(HY12), 1461–1476.

Ridolfi, L., Porporato, A., and Revelli, R. (2006). “Green’s function of thelinearized de Saint-Venant equations.” J. Eng. Mech., 10.1061/(ASCE)0733-9399(2006)132:2(125), 125–132.

Singh, V. P. (1994). “Accuracy of kinematic-wave and diffusion-waveapproximations for space-independent flows.” Hydrol. Process., 8(1),45–62.

Singh, V. P. (1996). Kinematic wave modeling in water resource-surfacewater hydrology. Wiley, New York.

Singh, V. P., and Aravamuthan, Y. (1995). “Errors of kinematic-wave anddiffusion-wave approximations for time-independent flows.” WaterResour. Manage., 9(3), 175–202.

Tsai, C. W.-S. (2003). “Applicability of kinematic, noninertia, and quasi-steady dynamic wave models to unsteady flow routing.” J. Hydrol.Eng., 10.1061/(ASCE)0733-9429(2003)129:8(613), 613–626.

Venutelli, M. (2011). “Analysis of dynamic wave model for unsteady flowin an open channel.” J. Hydrol. Eng., 10.1061/(ASCE)HY.1943-7900.0000405, 1072–1078.

JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / MAY 2014 / 965

J. Hydrol. Eng. 2014.19:956-965.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Uni

vers

ita S

tudi

Di P

isa/

Sba

on 0

4/28

/14.

Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.