Modeling debris flow propagation and deposition

Phys. Chetn. Earth (C), Vol. 26, No. 9, pp. 651-656,200l 0 2001 Elsevier Science Ltd.

Pergamon

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Modeling Debris Flow Propagation and Deposition P. Ghilardi’, L. Natale’ and F. Savi’ ‘Dipartimento ‘Dipartimento Received

di Ingegneria Indraulica e Ambientale, Via Ferrata 1, 27 100 Pavia, Italy di Idraulica, Trasporti e Strade, Via Eudossiana 20,OO 184 Roma, Italy

19 July 2000; accepted 12 April 2001

Abstract A mathematical model is applied to simulate two debris flows occurred in 1997 and 1998 in Italian Alpine valleys. The flows caused a casualty and significant damages to buildings and roads. The model considers a two-phase mixture of coarse sediments and interstitial fluid. The concentration of the finer solid fraction in the interstitial fluid is assumed to be negligible,. so that this fluid acts as clean water. Assuming that the solids and the interstitial fluid move downstream with the same velocity, the flow of the mixture is described using a two dimensional depth averaged model with a unique 2D momentum equation and two mass balance equations for the mixture and the sediments, respectively. Erosion and deposition rates are computed with a modified version of the relationship developed by Egashira-Ashida. Differential equations are integrated with an upwind explicit finite-difference scheme. The total discharge hydrograph at the upstream sections of the alluvial fans was estimated by assuming equilibrium solid concentration values. 0 2001 Elsevier Science Ltd. All rights

Many authors have proposed mathematical models of debris flow based on the conservation of mass and momentum of the flow. Only some of them take into account the erosion/deposition process and the different behaviour of different classes of sediment in the mixture. The fluid is alternatively considered as a one-phase constant-density fluid or a two-phases variable-density mixture composed by granular material immersed in an interstitial fluid. In this second approach two further conditions can be simulated: debris flow propagating on a fixed channel bed with no deposition allowed or on a bed of variable elevation due to erosion and deposition. The first assumption leads to a mathematical model which behaves very closely to a one-phase model. These hypotheses strongly influence the choice of the rheological model: the typical situation of a debris flow stopping where channel slope decreases may be simulated either with a constant density fluid or with a variable density mixture, but in the former cases the debris flow stops only if the rheological model allows for a yield stress. On the other hand, in a variable density mixture sediments may settle even when the interstitial fluid continues to flow downstream. For the constant-density fluid, or for fixed channel bed models, various rheological models where adopted such as, e.g., quadratic shear stress model (O’Brien et al, 1993) Bingham type model (Jan 1997, Fraccarollo 1995, Jin and Fread 1997) Herschel-Bulkley model (Laigle and Coussot, 1997; Coussot, 1997). Iverson (1997) introduces in his model the resistances due to Coulomb bed friction and shearing of the viscous pore fluid. Takahashi and Tsujimoto (1985) using a Bagnold rheological model in which the yield stress is not present simulated the entire phenomenon considering separate mechanisms for deceleration, stopping and deposition stages. Rickenmann and Koch (1997) tested different rheological models varying from Bingham to Newtonian fluid (both in laminar and turbulent flow), from dilatant to Voellmy fluid.

reserved

1. Introduction In mountain torrents, intense and localised storm may cause floods with important sediment transport. In steep torrents, the sediment discharge may increase so that the volumetric solid volumetric concentration often exceeds 40-50%. This is the case of debris flows that move huge volumes of sediments that are then deposed on the alluvial fans, often highly populated. These areas are periodically exposed to catastrophic events. To reduce debris flow hazard, it is common to couple structural and non structural protections, such as zoning of the risk prone areas and emergency plans. Protection plans require the definition of scenarios that can be defined by means of simulations with mathematical models. Correspondence

to: Paolo Ghilardi

651

652

I? Ghilardi et al.: Modeling Debris Flow

Of course, a constant density fluid model or a fixed bed model can not simulate the effects of sediment separation needed to reproduce those real world events in which the coarser sediments settle in the upper part of the alluvial fan or near obstacles. Modelling the fluid as a two-phases mixture capable to exchange material with the bed overcomes most of the limitations mentioned above, allowing for a wider choice of rheological models. Again, many alternatives can be found in the literature: for example, Bagnold’s dilatant fluid hypothesis (Shieh et al. 1996, Takahashi 1991, Takahashi et al. 1994), Chezy-type equation with constant value of the friction coefficient (Hirano et al. 1997; Armanini and Fraccarollo, 1997), cohesive yield stress (Egashira et al. 1997). Other rheological models were proposed for debris flow (see, e.g., Chen, 1988), and many of them can be easily used in numerical simulations; for a review see, e.g., Hutter et al. (1996). In this paper the debris flows occurred on 28 june 1997 in Rondinera and on 27 june 1998 in Ardenno, both located in Italian Alps (Northern Italy), are simulated with the computer code ALC0_2D developed by the authors. This code, which constitutes an evolution of the DBRFLW_2D code (Ghilardi et al, 1999), takes into account erosion and deposition by means of both the equations proposed by Egashira and Ashida (1987) and Takahashi (199 1); the aim of this study is to check the model capability to reproduce both the observed inundated area and the spatial distribution of deposits.

2 (ch)+ v

l

(cm) = ic,

where i is the bed erosion rate, c concentration and c, is defined as follows: c.

the

sediment

(i 2 0)

cu =max(c,c;,)(i <0)

(5)

i

where c* and cb are the solid concentrations

in the static

bed and just after deposition, respectively. The evolution of the bed level z is computed through the following expression: @cosS+i=O

.

at In this case study, erosion through modified versions Egashira-Ashida (I 987):

;

= Ktan(8,

and deposition are evaluated of the formulas given by

-se)

where:

2. Basic Equations A single momentum equation is written assuming the solid phase moving with the same velocity of the liquid phase. When Boussinesq coefficients are set equal to 1.0 the equation can be written in compact tensor notation as follows:

pS is the coarse grain density, p/ the interstitial fluid density, and 4 the internal friction angle (subscriptsf, U, and e stand for friction, velocity, and equilibrium, respectively). The empirical constant K is set equal to 0.1 following the work of Brufau et al. (1999).

3. Case studies 3.1. Ardenno debris flow - pgh(Vz -J)

=0

(1)

where U is the flow velocity, p the mixture density, h the flow depth, g the acceleration due to gravity, z the bed elevation, J the friction slope, 9 the angle between the bed and a horizontal plane: sin(S) = -(VaJ ,

(2)

The continuity equations for the total mixture and for the solid phase only are respectively:

$+V.(Uh)=i

On June 26 and 27, 1998, two successive and independent debris flows inundated the small town of Ardenno. The debris flows destroyed the check dam on the upper part of the alluvial fan and damaged several buildings and all of the bridges on the channels. Maximum deposition, with a thickness of about 1.0-l .5 m, occurred immediately upstream of the bridges, indicated as A and B in Fig. 1, where the inundated area is shown: deposition of sediments occurred in the shaded grey area, while the area bounded by the thick line indicates the zone where the debris flow propagated. The debris flow occurred on 26 June was simulated by the model.

F’.Ghilardi et al.: Modeling Debris Flow

653

Fig. 2 Ardenno debris flow: sediments upstream the upper bridge A

settled immediately

Sediment concentration in the upstream part of the alluvial fans was assumed to be equal to the equilibrium value:

c, = min

(8)

In this case study, c, = 0.9c* .

Fig. l- Ardenno

inundated

debris flow (June 1998): contour of areas; deposition areas are shaded.

Consequently, the upstream boundary equation:

r&d&.2The catchment area is 1.7 km2. A rainfall hyetograph was recorded by a gauge located very close to Ardenno; the average rainfall intensity was about 65 mm/h and the duration of the storm was about 12 minutes. A rainfallrunoff analysis, carried out by means of the Gamma unit hydrograph (Nash, 1960), allowed to estimate the liquid The runoff coefficient was discharge hydrographs. assumed equal to 0.7, assuming saturated soil immediately before the storm. The parameter of Gamma IUH were estimated according to MC Sparran (1968). No information concerning the sediment-grading curve, the density of the solid fraction, and the concentration of the static bed were available. We assumed: c*=O.65, &35” and p,=2650 kg/m3. The photo in Fig. 2, that refers to the upper bridge A, documents that a significant fraction of the sediments was constituted by boulders.

c* -cm

total discharge was estimated

hydrograph by means

at of

the the

(9)

where Q, is the computed water discharge, Q the mixture solid concentration discharge and ca, the equilibrium (Takahashi, 1991). The computational domain was discretised with 99x41 square cells with a size of 10 m; the DTM was obtained from a 1:2000 map and it was refined in order to take into account the top of levees, and the presence of houses and other buildings on the alluvial fan. In Figs. 3 and 4 the results of the model are compared with the observed data: Fig. 3 refers to the inundated areas, and Fig.4 to the deposition zones. The simulation results are sufficiently good excluding the southern part of the domain where the model simulates an overtop of the channel levees and a spreading of the wave in the alluvial fan: this overtop did not occur in the real world event.

P Ghilardi

654

et (11.: Modeling Debris Flow

Observed Computed

Fig. 3 - Comparison inundated zones.

between

observed

and computed

3.2. Rondinera debris flow On June 26, 1998, a landslide caused by an intense 20minute storm triggered a debris flow which propagated in a gully having an average bed slope of about 40%, with maximum values greater than 70%. The flow overtopped and partially destroyed a check dam before reaching the small town of Rondinera located on the upper part of the alluvial fan. The catastrophic event produced a casualty and several buildings and a road were damaged. The average thickness of sediment deposit was about 2 m, with maximum values of about 3-4 m. The volume of the moved solid material was about 24,000 m3. The size of the grains was strongly heterogeneous. Ten percent of grains had a diameter greater than 50 mm, and 20% a diameter lower than 0.1 mm. The average diameter was measured to be d=28 mm. The measured concentration of the static bed was c*=O.65. The measured static friction angle and the density of the solid particles were @35” and ps=2740 kglm3, respectively. In absence of any recorded

Fig. 4 deposition

Comparison zones

between

observed

and computed

rainfall hyetograph, a triangular discharge hydrograph with a peak discharge of the mixture flow Qd_max = 40 m3ls and a duration of 20 minutes was assumed at the upstream boundary, where uniform flow conditions were imposed. The value of the concentration was computed according to equation (8) and the result was cw=O.9c*. The flow started as supercritical. The computational domain was discretised with 81x57 square cells with a size of 5 m; the DTM was obtained from a 1:2000 map and was edited in order to take into account the top of levees, and the presence of houses and other buildings on the alluvial fan. In Fig. 5 the observed deposition area is compared to the computed one. Fig. 6 shows the evolution of the channel bed during the debris flow along the longitudinal section AB. The check dam upstream the town was filled in 4-5 minutes.

F?Ghilardi ef al.: Modeling Debris Flow 290 / -

655 ~.--.-..---. _...-..._ -t=O min ,’ . . . . t=4 min ‘i --

~~~~~~ t=IO min -- t=22 min

a 250 ; 240 S 230

Fig. 6 - Rondinera debris flow (June 1997): evolution of the channel bed along the longitudinal section A-B (see Fig. 5

k_ / \/

Computeddepositionarea Observed deposition area

010

50

1CQm

Fig. 5 - Rondinera debris flow (June 1997): comparison between observed and computed deposition areas.

4. Conclusions Two recent debris flows occurred in Northern Italy are simulated with a mathematical model taking into account sediment deposition and erosion and bed evolution. Computed results reproduce both the zone interested by the debris flow and the spatial distribution of sediments with a satisfying accuracy for the most part of the domain. It is confirmed that the debris flows propagation is strongly influenced by geometric terrain details that can hardly be identified on the available detailed maps: only direct field surveys allow to evaluate the elevation of the levees, and represent the road network precisely. These results suggest some important considerations for technicians dealing with the protection of the mountain areas from natural hazards. The debris flow volumes and discharges are much greater than the corresponding values of clean water, that are usually considered to design hydraulic structures like dams and bridges. In fact, the peak discharge of the mixture can be one order of magnitude greater than the liquid discharge due to rainfall. Moreover the density of the mixture can be two times greater than the density of the liquid flow and, consequently, the hydrodynamic forces acting on the

structures is increased. In alluvial fans, the direction of debris flows propagation can be estimated a priori with difficulty because the ground elevation changes during the passage of the debris flow, as occurred in Rondinera. Usually the hydraulic structures are designed referring to the peak discharge and the volume is considered a variable of minor importance: however, the application of mathematical models shows that the propagation and the stoppage of the debris flow are significantly affected by the volume of the mixture. We can conclude that hydraulic structures located in debris flow risk prone areas that are designed taking into account liquid discharge only should be considered unsafe. Acknowledgemenis

The authors kindly acknowledge Albert0 Fioroni for help in field measurements and data processing. This work was partially granted by program “Morfodinamica Fluviale e Costiera” of the Ministry of University (MURST) and by program “Mappatura del rischio idraulico in aree di conoide” of Regione Lombardia.

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Laigle, D. and Coussot, P., Numerical modelling of mudfows, Journal of Hydraulic Engineering., Vol. 123, NO. 7.617-623, 1997. MC Sparran, J..E., “Design hydrographs for Pennsylvania watersheds”, Journal of Hydraulic Division, Proc. Am. Sot. Civ. Eng. 94, HYI, 1968. Nash, E.J., A unit hydrograph study, with particular reference to Brilish catchmenrs, Proc. Institution of Civil Engineers, 17, 1960. O’Brien, J.S., Julien, P.J., Fullerton, W.T., Two-dimensional water flood and mud/Tow simzdarion, Journal of Hydraulic Engineering., 119 (2). 1993. Rickenmann, D and Koch, T., Comparison of Debris Flow Modeling Approaches, in Cheng-Lung Chen (Ed.), Debris flow hazards mitigation: mechanics, prediction an assessment, ASCE, 1997. Shih B.J., Shieh CL. and Chen L.J., (1997), The grading of risk for hazardous debris flow zones”, in Cheng-Lung Chen (Ed.), Debris Bow hazards mitigation: mechanics, prediction an assessment, ASCE, 1997 Takahashi, T., Debris Flow, Balkema, Rotterdam, 1991. Takahashi, T. and Nakagawa, H., Flood/debrisJlow hydrograph due to collapse of a natural dam by overtopping”, Journal of Hydroscience and Hydraulic Engineering, 1994. Takahashi, T. and Tsujimoto, H., Delineation of fhe debrisf7ow hazardous zone by a numerical simulation method, int. Symp. on Erosion, Debris Flow and Disaster Prevention, 1985.