1d mathematical modelling of debris flow

7/27/2019 1D Mathematical Modelling of Debris Flow

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JOURNAL OF HYDRAULIC RESEARCH, VOL. 38, 2000, NO. 6 435

Revision received July, 1999. Open for discussion till June 30, 2001.

1D Mathematical modelling of debris flow

Simulation numrique unidimensionnelle du charriage torrentiel

P. BRUFAU, P. GARCA-NAVARRO,Mecnica de Fluidos. Centro Politcnico Superior. Universidad de Zaragoza. Spain.

P. GHILARDI, L. NATALE,Dipartamento di Ingegneria Idraulica e Ambientale. Universit degli studi di Pavia. Italy.

F. SAVI,Dipartamento di Idraulica Transporti e Strade. Universit La Sapienza di Roma. Italy.

ABSTRACT

Debris flow is modelled using the equations governing the dynamics of a liquid-solid mixture. An upwind finite volume scheme is applied to solvethe resulting differential equations in one dimension. These equations have a structure similar to those of the monophasic water flow, differing fromthem by the presence of some terms characteristic of the bifasic nature of the mixture, such as granular bed erosion velocity, sediment concentration,bed shear stress, etc. The model and the system of equations to be solved are presented with the description of the implementation of the upwindscheme for the resulting hyperbolic conservation system. The numerical method is first order in both space and time. The treatment of the sourceterms is specified in detail and some comparison with laboratory experiments are presented.

RSUM

Le charriage torrentiel est represent par un systme dquations unidimensionnelles dcrivant le comportement dun mlange liquide-solide. Lamthode numrique utilise est une mthode aux volumes finis, utilisant une discrtisation de type upwind. Les quations sont similaires dans leur

forme celles dun coulement surface libre monophasique, mis part la prsence de quelques termes associs la nature diphasique du mlange,tels que la vitesse derosion du lit granulaire, la concentration en sdiments, le cisaillement sur le fond, etc. Le systme dquations hyperboliques etla mthode numrique sont prsents. Cette mthode numrique est du premier ordre en espace et en temps. Le traitement des termes source faitlobjet dune desription dtaille et lon prsente des comparaisons avec des expriences en laboratoire.

1 Introduction

In mountain torrents, intense and localised storms may causeflash floods with important sediment transport. In steep tor-rents, the sediment discharge may increase so that the solid con-centration often exceeds figures of 40-50%. This is the case ofthe debris flows that transport downstream huge volumes ofsediments that are then deposited on the alluvial fans, oftenhighly populated. These wide areas are periodically exposed tocatastrophic events. To reduce the debris flow hazard, it is com-mon to couple structural and non structural protections such aszoning of the risk prone areas and emergency plans. Protectionplans require the description of scenarios that can be definedonly by means of simulations with mathematical models.In this paper a numerical model for unsteady debris flow is pre-sented. The model will be applied to simulate different test con-ditions in channels with simple geometry. A one-dimensionalscheme is proposed whilst a two-dimensional scheme for themore complex wave propagation on alluvial fans will be devel-

oped in future work.Many authors have proposed mathematical models of debrisflow based on the conservation of mass and momentum of theflow. Only some of them take into account the erosion/deposi-tion process and the behaviour of different classes of sedimentin the flow. The fluid is alternatively considered as a one-phaseconstant-density fluid or a two-phase variable-density mixturecomposed by granular material immersed in an interstitial fluid.This assumption strongly influences the choice of the rheologi-cal model: the typical situation of a debris flow stopping wherethe channel slope decreases may be simulated either with a con-stant density fluid or with a variable density mixture; but in the

former case, the debris flow stops only if the rheological model

allows for a yield stress. On the other hand, in a variable densitymixture, the sediments settle even though the interstitial fluidcontinues to flow downstream.For the constant-density fluid, different rheological modelswere adopted in the past such as Bingham type model (Fracca-rollo 1995, Jan 1997, Jin and Fread 1997), Herschel-Bulkleymodel (Laigle and Coussot 1997) or quadratic shear stress

model (O'Brien et al. 1993). Takahashi and Tsujimoto (1985),using a Bagnold rheological model in which the yield stress isnot present, simulated the entire phenomenon considering sepa-rate mechanisms for deceleration, stopping and depositionstages. Rickenmann and Koch (1997) tested different rheologi-cal models varying from Bingham to Newtonian fluid (both inlaminar and turbulent flow) and from dilatant to Voellmy fluid.A constant density fluid model cannot simulate the effects ofsediments separation, needed to reproduce those real worldevents in which the coarser sediments settle in the upper part ofthe alluvial fan or near obstacles in the river bed.Modelling the fluid as a two-phase mixture overcomes most of

the limitations mentioned above, allowing for a wider choice ofrheological models. Again, many alternatives can be found inthe literature, for example: Bagnold's dilatant fluid hypothesisused by Takahashi (1991), Takahashi and Nakagawa (1994),Shieh et al. (1996); Chezy-type equation with constant value ofthe friction coefficient (Hirano et al. 1997, Armanini and Frac-carollo 1997); cohesive yield stress (Egashira et al. 1997), etc.Other rheological models were proposed for debris flow (seeChen 1988), and many of them can be easily used in numericalmodelling. For a recent review see Hutter et al. 1996.The change in the debris flow density can be modelled through

the mass balance of both phases (solid and liquid), and the defi-nition of the erosion/deposition rate as a function of sediments


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concentration. Shieh et al. (1996) introduced an empirical non-equilibrium condition for concentration when depositionoccurs, assuming that concentration varies from the equilibriumin the steeper reach to the equilibrium concentration in the flat-ter reach according to an exponential law; Armanini and Frac-carollo (1997) assumed the concentration equal to theequilibrium value. Egashira and Ashida (1987), Hirano et al.

(1997), Honda and Egashira (1997), Mizuyama and Yazawa(1987) and Takahashi et al. (1987) developed 1D and 2D mod-els which consider non-equilibrium conditions. The first threeauthors take only into account the coarse fraction, i.e. the inter-stitial fluid is nearly homogenous (water). The last two authorsconsider also variations of the concentration of the fine fractionin the interstitial fluid.The erosion/deposition rate is controlled by the excess of thelocal instantaneous concentration over the equilibrium concen-tration. Egashira and Ashida (1987) and Honda and Egashira(1997) computed this rate by means of a simple relationship.Takahashi (1991) proposed semi-empirical expressions whichdiffer from deposition to erosion and erosion produced in a sat-urated bed from that in an unsaturated bed is distinguised too.All these models ignore the spatial and temporal variations ofdebris flow density in the momentum balance equations.Most of the above models are solved numerically with finitedifference schemes. Takahashi and Tsujimoto (1985) used acentred differences scheme; Fraccarollo and Toro used anupwind scheme proposed by Toro (1996); Honda and Egashira(1997) proposed a technique based on backward differences,valid only for supercritical flow; other authors used Lagrangianschemes (see Savage and Hutter 1989, Iverson 1997, Ricken-

mann and Koch 1997).In the present work granular and liquid phases are considered.The set of equations includes two mass conservation equations(one for the mixture and another for the solid phase) and a sin-gle momentum balance equation of the 1D flow. The spatial andtemporal variation of debris flow density is included as a sourceterm. The friction term is simulated according to Takahashi(1991). The system is completed with equations to estimate theerosion/deposition rate derived from the Egashira and Ashida(1987) or Takahashi (1991) relationships.The set of equations is solved by means of an explicit finite vol-ume technique based on first order Roe's scheme. The advection

equation of the coarse solid fraction is solved in cascade at eachtime step after the momentum balance equation of the mixturehas been integrated.

2 Governing equations

The two-phase mixture constituted by coarse sediment fractionand interstitial fluid is considered. The concentration of thefiner solid fraction in the interstitial fluid is assumed to be neg-ligible, so that the fluid acts as clean water. The same velocityfor the coarser solid fraction and the interstitial fluid is assumedtoo, therefore a unique momentum equation is used. The flowof the solid-liquid mixture is described using a 1D depth aver-aged model that, apart from stating mass and momentum con-

servation of the debris, includes a solid phase massconservation law and a bed evolution law. The set of four dif-ferential equations is presented in this section.The equations describing the mixture and the coarse fractionmass balance and the mixture momentum along the mainstream direction can be expressed in

Mass and momentum balance equations for the mixture solid-liquid can be expressed in conservative form as (Chow 1959)

with

In (2.4) U

represents the vector of conservative variables, F

isthe flux vector in the x

direction, and H

is the source term. Inaddition, A

is the flow cross section area, Q

is the flow dis-charge, h

is the depth of the debris flow, u

is the mean velocityin a cross section, c

is the dimensionless volumetric concentra-tion of sediments in the mixture, is a parameter related tothe bed concentration of sediments to be defined later, b

is thewidth of the channel, g

is the acceleration due to gravity, i

is thebed erosion/deposition velocity and

is the momentum correc-tion coefficient that we will assume to take the value

= 1 fromnow on. The bed slope is given by the bed inclination

wherez

is the bed level respect to an arbitrary horizontal refer-ence. In the mathematical model we are presenting, the bedlevel may change in space and time so that definition of the bedevolutionz = z(x,t)

is one of the objectives. These changes arerepresented as erosion and/or deposition phenomena. We haveonly modelled bottom movement in all the cases, avoiding thepossible change of the channel vertical walls, as a preliminarwork to check the capability of the model. On the other hand,

rectangular cross sections have been assumed, although infuture work arbitrary geometries for the cross sections with

At------ Q

x------- ib=+ (2.1)

Qt-------

Q2

A---------- g

A2

2b------ cos+

x----------------------------------------------- gA S0 Sf( )=+ (2.2)

hc( )t

-------------- hcu( )x

----------------- ic v=+ (2.3)

Ut------- F

x------ H=+ (2.4)

U AQ

FQ

Q2

A------ g

A2

2b------ cos+

Hib

gA So Sf( ) =,=,= (2.4.1)

cv

So

So sinzx------= = (2.4.2)


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437

their complete change in time and space will be considered aswell as 2D problems.For a detailed review of different equations describing the fric-tion term, represented by , see Chen (1988). In the presentwork, the Takahashi (1991) equations have been chosen accord-ing to the dilatant fluid hypothesis developed by Bagnold(1954). Two main different types of flow have been distin-

guised: For low concentrations of the coarse fraction in water (

c

0.2),friction is modelled by a typical open channel law and the flowis classified as immature

debris flow. The Mannings equationfor friction used in open channel free surface flows (Chow1959), is replaced by

In this law, d

is the mean effective diameter of the sedimentparticles and is the hydraulic radius where P

is the wetted perimeter. For rectangular channels

P = b + 2h

. More complicated dependences appear when the concentra-

tion of solid is important (

c

0.2). In that case the flow isclassified as stony

debris flow and the law suggested byTakahashi (1991) is

An explicit dependence on the concentration is shown in(2.6). Moreover, the quantity , linear concentration,depends on the granulometry of the solids in the form

where c*

is the coarse fraction concentration in the staticdebris bed, a

B

is an empirical constant (Bagnold (1954)assumed a

B = 0.042),

is the dynamic internal angle of fric-tion, and are the sediment and water density respec-tively.

These resistance equations were validated through experimentsin flumes with almost smooth and fixed walls. In presence of astatic granular bed, the roughness is significantly higher. Taka-hashi (1991) fitted his experimental data in flumes with almostsmooth and fixed walls using for a

B

the value given by Bagnold

a

B

= 0.042. In presence of an erodible granular bed, he foundsignificantly higher resistance so the value ofa

B

was increasedto 0.35-0.5.When the sediment concentration is high, the resistance ismainly caused by the dispersive stress and the roughness of thebed does not influence the resistance (Scotton and Armanini1992). For low values of the sediment concentration or ,

the energy dissipation is mainly due to the turbulence in theinterstitial fluid and the influence of the wall roughness

becomes important. In such case, Takahashi (1991) suggests theuse of Mannings equation or similar resistance laws.It is worth noting that density is not present in these equationsas a variable. The underlying assumption is that, being differentfrom water density, there is a uniform and constant debris den-sity. This will be reconsidered at the end of this section.The mass balance equation in (2.1) expresses actually volume

conservation of the mixture. Therefore, the bed erosion/deposi-tion velocity

i appears as a source term. The dependence of thisquantity with the basic set of dependent variables has to bemodelled. In this work two models have been chosen; a simpleone proposed by Egashira and Ashida (1987) in the form

involving the difference between the bed slope and theequilibrium angle . The empirical coefficient K

is assumedequal to 1 by Egashira and Ashida.The equilibrium angle is a relevant parameter that depends

mainly on the concentration of the mixture and on the densityratio between solid and water. When the inclinationof the channel bed has reached the equilibrium angle, no ero-sion or deposition occurs and a steady bottom state can bedefined. This variable will determine the test cases to check thescheme.

is the static internal friction angle and c

is the sediment volu-metric concentration. A positive value of i

means that erosiontakes place, otherwise deposition occurs until equilibrium isreached.In case a debris front advances over an adverse slope,

and equation (2.8) will always tend to deposition even whenerosion should occur. In order to overcome this difficulty amodification of the equation is proposed in this work

where

being S

f

the slope of the energy line. When there is a uniformflow, , and ifK = 1, equation (2.11) becomes equal tothe original expression proposed by Egashira and Ashida (2.8).Although (2.11) is introduced as an alternative expression of

(2.8) for adverse slope reaches, it should be noted that (2.11) is

Sf

Sfd

2u

2

0.49gh 2Rh-------------------------= (2.5)

Rh Rh A P=

Sfu

2

25d------

1---h

1aB sin-----------------

2

c 1 c( )ls----+ gRh

-------------------------------------------------------------------------------------= (2.6)

c*

c----

13---

1

1

= (2.7)

s, l

h

d--- 30>

i Ku e( )tan= (2.8)

( )e( )

s s l=

e arc c s 1( )

c s 1( ) 1+----------------------------tan

tan= (2.9)

zx------ 0 S0 0 0 i 0 (2.10)

i Ku f e( )tan= (2.11)

f arcSf

cos------------

tan= (2.12)

f =


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valid also for the entire channel independently of the sign of thebottom slope.The original model from Egashira and Ashida (2.8) is based onthe assumption of normal flow so driven by the bed slope.Instead, (2.11) implies that the erosion/deposition is forced bythe energy line slope and is associated to a slower process oferosion/deposition in general, shown in Fig. 2.

Modifying the Takahashi's equations (1991), Ghilardi andNatale (1998) derived the following expressions for the erosion/deposition rate. Takahashi (1991) distinguises when depositionor erosion take place; if erosion occurs (c > c'eq) in presence ofa saturated bed, i is calculated in the form

where is the equilibrium concentration for non-uniformflow

and represents the equilibrium concentration for uniformflow

being an empirical constant adjusted with experimentalmeasurements. Takahashi proposed for erosion,and in case equilibrium state is predicted, it seems to agree withphysical phenomena. In order to fit the experimental resultsgiven by Gregoretti (1996), the value of has been increasedone order of magnitude, .Otherwise, when deposition occurs, the model adopted for thedeposition velocity is

with , an empirical constant for deposition as wellas for erosion. In the tests presented, the value chosen is

.For physical reasons, , being the value of theempirical constant m = 0.9 (Takahashi 1991) or 0.95 (Yazawaand Mizuyama 1987), derived from experimental observations.Since in real world cases the maximum values of and

are often reached, the value of m strongly influences theresults.The equation for the sediment mass conservation, (2.3), is againestablished as a volume balance and repeated here for completi-tude

cv is a parameter of the model related to the concentration ofsediments in the bed layer.

where is the volumetric sediment concentration on a static

bed just after deposition.Finally, when the cross section of the channel is rectangularwith fixed walls and loose bottom, the movement of the bed dueto erosion or deposition processes that take place in presence ofa given sediment concentration is represented by the followingequation

The complete set of partial differential equations to solve is

for the variables A, Q, z, c. They form a closed set if (2.5) or(2.6), (2.11) or (2.13) and (2.16) are used.Another form of the equations has been adopted to permit den-sity changes in time and space in a suitable and simple way. Inthis case equation (2.2) has been replaced by

The debris density can be expressed in terms of the concen-tration , the solid density and the liquid density ,

i eceq cc* ceq-----------------

tan ftantan tan

------------------------------- hud

------= (2.13)

ce q

ce qftan

s 1( ) tan ftan( )----------------------------------------------------= (2.14)

ceq

ceqtan

s 1( ) tan tan( )--------------------------------------------------= (2.15)

ee 7 10

4=

ee 7 10

3=

i dceq c

c*-----------------u= (2.16)

0.05 d 1 e

d 0.05=ceq ceq mc *,

ceqceq

hc( )t--------------

hcu( )x----------------- ic v=+ (2.17)

cvmax c cD

*,( ) i 0

c* i 0>

= (2.18)

cD*

zt----- cos i 0=+ (2.19)

At------ Q

x------- ib=+ (2.20)

Qt-------

Q2

A------ g

A2

2b------ cos+

x-------------------------------------------- gA S0 Sf( )=+ (2.21)

hc( )t

-------------- hcu( )x

----------------- ic v=+ (2.22)

zt----- cos i 0=+ (2.23)

Q( )t---------------- Q

2

A

------ gA2

2b------ cos+

x--------------------------------------------------- gA S0 Sf( )=+ (2.24)

c s l

sc l 1 c( )+= (2.25)


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The rest of equations remain the same. With algebraic opera-tions, isolating the terms that contain and using the continuityequation we arrive to

where

is the density of static bed and the density of static bedjust after deposition. The new term in the right hand side con-taining density variations has been treated as a source term and

has been discretized explicitly like the other source terms.Comparisons of these two models for the momentum equationhave been carried out in all the numerical test cases withoutshowing any appreciable difference in the results. The debrisdensity variations are not strong enough to change the value ofthe variables. This confirms the hypothesis followed by severalauthors (Mizuyama and Yazawa 1987, Nakagawa and Taka-hashi 1997), who ignore the effects of the spatial and temporalvariations of density in the momentum balance equation.

3 Numerical model

The domain where the variables are going to be calculated isdivided in a temporal-spatial computational mesh. Each point isrepresented by the pair , wherexj represents the positionin the spacej = 1, 2, ....,JMAXand tn the time level n = 1, 2, ....,

N.Upwind schemes are based on discretizations depending on theform of the equation and on the solution itself, taking intoaccount the physical influence domain of every point in themesh. They have proved well suited for the numerical solutionof conservation laws.In an explicit form, the first order upwind scheme for equation

(2.4) can be written as:

This cell-wise formulation is based on a balance over cellj withand . In this work, a

uniform mesh has been chosen in space.is the numerical flux defined by

and is based on Roe's approximated Jacobian of the flux (Roe1981).The construction of the upwind scheme goes through a locallinearization of the system of equations

with

is the celerity of the linear surface waves. Anew, approximated matrix is defined with the followingproperties

,

,

has real and distinct eigenvalues and a completeset of eigenvectors and

.

Roe (1981) proposed for the Jacobian matrix of theexact flux evaluated in an average state A . Therefore,the problem is moved to the computation of this averaged state,which is obtained imposing property (2).

Since the eigenvalues and eigenvectors of the matrix are:

Average values and are used to define similareigenvalues and eigenvectors of the approximated matrix

:

From property (2) and the conditionis easy to deduce that

(Alcrudo and Garca-Navarro 1992)

Qt-------

Q2

A------ g

A2

2b------ cos+

x-------------------------------------------- gA S0 Sf( )=+

bu i

-------- v ( )

g

---A

2

2b------

x------cos+

(2.26)

vD

* i 0

*i 0>

= (2.27)

* D*

xj tn,( )

Ujn 1+ Uj

n tx

------ Fj

12---+

* Fj

12---

* n t Hj

n+= (3.1)

x xj 12---+ xj 12---

= xj 12---+

12--- xj xj 1++( )=

F*j

12---+

F*j

12---+

12--- Fj 1+ Fj Aj 1

2---+

Uj 1+ Uj( )+= (3.2)

Ut------- A

Ux------- H=+ (3.3)

A FU-------

0 1

v2

u2 2u

= = (3.4)

v gh cos=Aj 1 2( )+

1) Aj

1

2---+

Aj

1

2---+

Uj 1+ Uj,( )=

2) Fj 1+ Fj Aj 12---+

Uj 1+ Uj( )=

3) Aj 1/2+

4) Aj 1/2+ Uj Uj,( ) A Uj( )=

Aj 1/2+Uj 1/2+ )(

Aa

1 2,u v= (3.5)

e1 2, 1

a1 2,

= (3.6)

uj 1/2+ vj 1/2+

A Uj 1/2+( )

aj

12---+

1 2,u

j12---+

vj

12---+

= (3.7)

ej

12---+

1 2,1

aj

12---+

1 2,

= (3.8)

Uj 1+ Uj k 1 2,= j

12---+

k ej

12---+

k=

uj 12---+

hj 1+ uj 1+ hjuj+

hj 1+ hj+---------------------------------------------= (3.9)


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Then, the numerical flux corresponding to the first orderupwind scheme given in (3.2) can be written as

In order to avoid the problem of numerical solutions with non-physical discontinuities (zero eigenvalues), incompatible withthe entropy principle, Harten and Hyman (1983) proposed theredefinition of the absolute value of the eigenvalues of .It is necessary on this purpose to define the quantity

The new absolute values of the eigenvalues are defined as

The source terms have been discretized pointwise in space andexplicitly in time, except the friction term that has been treatedin an implicit way to improve numerical stability. At the begin-ning of this work great difficulties with source terms arose. Theflow advance over an initially dry bed and huge friction termsproduced oscillations and finally the explosion of the code. Thisproblem was solved evaluating the discharge in friction term attime level n+1.For example, in equation (2.5) the new friction term consideredis

In this case, the discretized equation (3.1) for the momentumequation (2.2), is replaced by the following second order equa-tion

where

and

Equation (3.16) is solved as an ordinary second order equationevaluating before the sign that the discharge variable shouldhave.

where

because physically the friction term is not capable of changingthe sign of the discharge.Equation (2.3) is discretized in the same form as (2.4), althoughit is a scalar non-linear equation instead of a non-linear systemof equations. The equation can be written in the same form as(2.4)

where

Using the conservative formulation, an upwind scheme can beapplied in the form we have described before being the updat-ing at nodej performed as

Now,

and .

Finally, the discretization of equation (2.19) has been doneexplicitly, and the unknownz is calculated in the form

vj

12---+

ghj 1+ cos j 1+ hj cos j+

2-------------------------------------------------------= (3.10)

j

12---+

1 2, hj 1+ hj2

---------------------=1

2vj

1

2

---+

------------- hu( )j 1+

hu( )j uj

12---+

hj 1+ hj( )(3.11)

F*j

12---+

12--- Fj 1+ Fj a

j12---+

k j

12---+

k ej

12---+

k

k 1 2,=+= (3.12)

Aj 1/2+

j

12---+

k

j

12---+

kmax 0, a

j12---+

kaj

k aj 1+

ka

j12---+

k ,= k 1 2,= (3.13)

j

12---+

k

aj

12---+

k if aj

12---+

k j

12---+

k

j

1

2---+

k if aj

1

2---+

k j

1

2---+

k

1d mathematical modelling of debris flow

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