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A new method for debris-flow hazard assessment in alluvial fans

C. Ceccarelli, F. Napolitano & F. Savi Department of Hydraulics, Transportation and Highways, University of Rome "La Sapienza "

Abstract

A new method to identify the probability of an area in an alluvial fan being flooded is proposed. This method takes into account in a simplified way the propagation of debris flow in the main channel, evaluate the outflowing discharge and the spreading of the outflow in the alluvial fan. The method is applied to identify the zones with the same probability to be flooded in the alluvial fan of Gianico, located on Italian Alps in Northen Italy. It allows to take into account the uncertainties in the estimation of the model parameters and the effects of these uncertainties in the extension of the risk- prone areas are discussed.

1 Introduction

Flood damages D are usually defined ([l], [2] , [ 3 ] ) as

where L is the value of the risk prone structures, normally expressed in monetary values, and V is the vulnerability that indicates the attitude of the structures to be damaged; V is dimensionless and varies from 0, no damages, to 1, total destruction. The total damage over a period o f t years, when N, floods occur, can be defined as:

N.

Transactions on Ecology and the Environment vol 60, © 2003 WIT Press, www.witpress.com, ISSN 1743-3541

352 River Ba~itl Marragernctlr I 1

The risk R over t years, is defined as the expected value of D,:

where E () is the expected value of the random variable in parenthesis. With some manipulations ([4], [ 5 ] ) the annual risk in a given area can be expressed as:

1 where T is the return period and, consequently, P = - is the probability of this

T area being flooded. Eqn (4) allows to frame in a rational scheme the measures to cope with floods. The procedure usually adopted to map the flood-prone areas involves the evaluation of the peak discharge for defined values of the return period and the computation of the corresponding flow depths [6] . This approach identify the areas that can be flooded for the considered values of the return period, but does not allow to evaluate the probability P of a given area in the alluvial fan being flooded. This goal can be achieved only by a stochastic approach; the probability P can be computed as:

m

where f is probability density function (pdf) of the peak discharge Q at the apex of the alluvial fan, and P* is the probability of the considered point being flooded if the peak discharge is Q. To solve eqn ( 3 , FEMA [7] introduced some assumptions on the geomorphology of the alluvial fan and on debris flow propagation that in real world applications are frequently not verified [8]. More specifically, 'FEMA procedure assumes that the channel that conveys flow across an alluvial fan has a random location, and the flow is confined to a portion of the fan surface which conveys debris flow from the apex to the toe. The shape of the channel is considered rectangular, and the flow propagates in critical conditions. The probability of a given point on the fan being flooded during a flood event decreases from the apex to the toe of the fan because of the widening or expansion of the fan surface in downslope direction. For an exhaustive discussion of the assumptions of the FEMA procedure and some modifications to the FEMA methodology, see French ([g]; [g]). Some assumptions result arbitrary in real world applications. If the alluvial fan is strongly populated, this is the case of most Italian Alpine valleys, frequently an entrenched channel exists at the apex of the fan and the direction of the channel differs from the direction of the median radial line. Moreover, not always the


alluvial fan contour width increases from the apex to the toe of the fan due to obstacles (structures, embankments, levees) which can reduce the extension of the area that can be flooded in downslope direction. To overcome this limitations, the proposed method solves the above integral via numerical integration, having estimated the function f by means of statistical inference; for each value of Q, increased to take into account sediment transport, the flow stages in the channel along the alluvial fan are evaluated by integrating the 1D steady equation of motion of a free-surface flow, and the outflowing discharges where the flow overtops the levees are computed. Next, the area being flooded is contoured by applying the procedure proposed by the Disaster Prevention Research Institute of Kyoto University [l01 and generalized by Ghilardi [ll]. The value of the probability P* is evaluated for each point of the alluvial fan (P* =l if the point is included in the flooded area, P*=O otherwise).

2 Description of the proposed method The alluvial fan constitutes the computational domain, which is discretised by means of a rectangular grid. In each point of the computational grid, the integral is solved via numerical integration as:

N

P(x>Y) = Cfj(Qj)J"~Qj (6) j=l

The summation in (6) is extended until the values of P in each point of the grid varies of less than 10.~. To solve eqn (6), the evaluation of the pdf f of the peak discharge Q at the apex of the alluvial fan is required. Usually, measurements of discharges in stream gauging stations are not available in alluvial fans; consequently the discharges are estimated from precipitations by means of a rainfall-runoff analysis. The total discharge at the apex of the alluvial fan is estimated from the water discharge Q!, computed by means of the rational method starting from the rainfall intensity- duration curve, according to the following equations:

where C is the volumetric solid concentration and C* is the maximum volumetric solid concentration. We assume that equilibrium conditions are reached at the apex of the alluvial fan, so that

tan B C =

(Ps - P) / p(tand - tan Q)

where tan B is the bed inclination at the apex of the alluvial fan, $ is the static internal friction angle, p and p, are the liquid and sediments density, respectively.


354 River B a ~ i t l Marragernctlr I 1

The probability P* of a given point being flooded if the peak discharge is Q is evaluated in two steps. Firstly, the flow profiles in the main channel are evaluated by integrating the steady energy and mass balance equations of 1D free surface flow [l21 for each values of the peak discharge Q:

where X is spatial coordinate measured along axis of the channel form the apex of the fan, q is the lateral outflowing unit discharge due to the overtopping of the levees, H is the total head and Sfis the slope friction. According to many Authors 1131, [14], [l 51, the fluid in eqn (9) is schematised as a one-phase constant-density fluid, instead of a two-phase variable-density mixture [16], [17]. The values of q are computed according to:

where Y is the water elevation in the channel, Z is the elevation of the top of the levees, p is the discharge coefficient and g is the gravity constant. Different rheological models are proposed to evaluate the slope friction, such as Bingham type model [14], Herschel Bulkley [15], quadratic shear stress model [13], and Bagnold's dilatant fluid [16]. In this paper the relationship found out by Rickenmann and Weber ([18]) on the basis of debris flow experiments in flume is used:

where U is the mean debris flow velocity, and

U - gRS , R is the hydraulic radius and g *-4 U * is the shear velocity

is the acceleration due to

gravity. Eqs (9) and (10) are solved by means of the standard step methods [l21 and the ouflowing discharge are evaluated. The second step involves the contouring of the areas that can be inundated. The values of the probability P* in eqn (5) is set equal to 1 if the considered point is included in the flooded area, and P*=O otherwise. To identify the area that can be flooded, the procedure proposed by the Disaster Prevention Research Institute of Kyoto (DPRI) [10], and generalized by Ghilardi [ l l] is adopted. According to Fig. 1, the extension of the deposition XL in the flow direction can be computed with


where: Cg cos sn tan 4

G = --p)+- gsin*n

and h is the debris flow depth.

Figure l : Longitudinal profile of debris flow during stoppage and identification of the inundated area

The DPRI procedure suggests to correct the value of the distance XL computed by means of eqn (12) in order to take into account the debris flow volume. No corrections are introduced in the proposed method. The values of the mean velocity and debris flow depth U, and h, are assumed equal to the corresponding critical-state values computed at the top of the levees, considered as weirs. Summarising, the proposed procedure involves the following steps to solve eq.n(6) for each point of the computational grid: a) a value of the solid discharge Q, is imposed at the apex of the alluvial fan, b) the value ofJ(Qj) is evaluated, c) the steady state profile in the main channel are evaluated by integrating

eqs (9, 10) and the values of the outflowing discharge Q,, are computed, d) the flooded area is defined according to eqn (12) and the graphical

procedure in Fig. l b, e) if the considered point of the computational grid is included in the

flooded area P*,=l, and if the point is outside the flooded area P*]=O, Steps from a) to e) are repeated for increasing values of the total discharge until the increment of the value of P(x,y) is lower than I O - ~ .



2.1 Evaluation of the uncertainties in the estimation of the parameters

The results of the proposed method are affected by two main sources of errors. The mathematical model (DPRI method) used to identify in a 2D domain the zones that can be flooded may constitute a rough schematisation of the debris flow propagation on an alluvial fan, and the estimated values of the parameters of the models (C*, p,, p) are strongly uncertain, due to lacking of direct field surveys. For these reasons the parameters of the model must be considered as random variable. Laigle and Marchi [l91 proposed an example of debris flow hazard assessment by defining some scenarios which include variations of the values of discharge, volumes and parameters. However the Authors did not introduce the probability distribution of the parameters. In this paper the effects of this second source of uncertainties in the results of the method are investigated following a Monte Carlo procedure. According to this statistical sample-analysis method, for any value of the peak discharge at the apex of the alluvial fan, step (1) in section 2, a sample of values of the model parameters are randomly generated from their probability distribution laws, which we suppose to know. We assume that each values of the model parameters are distributed according to a triangular probability distribution function. We adopted this very simple probability distribution function because frequently we have information about the mean value and about the limit values of each parameter. Of course, the choice of the probability distribution law does non affects the proposed method. Talung into account uncertainties in the values of the parameters, the proposed method is modified by introducing further steps: a) a value of the solid discharge Qj is imposed at the apex of the alluvial fan, b) the value of4 is evaluated, c) N scenarios of the values of the parameters are generated, d) for each scenario, the steady state profile in the main channel are

evaluated by integrating eqs (9, 10) and N values of the outflowing k discharge Qoj are computed ( k l , . . ., N),

e) for each scenario, the flooded area is defined according to eqn (12) and the graphical procedure in Fig. lb,

f) for each point of the computational grid the probability P ) is evaluated as P*,-MIN where M is the number of scenarios for which the considered point is flooded,


3 Real world application of the proposed method

The proposed method was applied to assess the probability of inundation of the Gianico alluvial fan, located in an Alpine region in Northern Italy (Fig. 2). A large portion of the alluvial fan was inundated in the past, especially the small town of Fucine, when severe floods propagate along the Re torrent; the more recent floods occurred in l966 and l98 1 [20]. The bottom channel slope of the Re torrent immediately upstream the apex of the alluvial fan is So=0.14; the mean slope of the bottom channel in the alluvial fan is So=0.07, varying Erom So=0.13, So=0.06. The mean width of the channel is 10 m. The total discharges at the apex of the alluvial fan are Q=86 m3/s and 109 m3/s for T=50 and T=100 years, respectively.

Figure 2: Gianico alluvial fan and Re torrent

The areas subjected to flooding with P-0.01 (T=100 years) obtained by applying the FEMA procedure are shown in Fig. 3; the values of the debris flow depth are also reported in the same figure. These areas are located in the riverine zone of Re torrent and in the lower part of the alluvial fan; the buildings in the town of Gianico close to the Re torrent resulted subjected to flooding, but the FEMA



procedure does not identify as subjected to flooding the town of Fucine. These results are obtained by adopting the GEV distribution [21] to define the discharge frequency curve and agree very closely with those obtained by Natale et al. 1221 that used the lognormal distribution. The areas that can be flooded by debris flow depth lower than 0.45 m can not be identified because some road embankments reduce the contour width in the lower portion of the alluvial fan, and in this case FEMA procedure can not be applied. The proposed method is applied and the areas subjected to flooding for different values of P=0.020, 0.010 and 0.005 are reported in Fig. 4. The main channel of Re torrent was discretised in 50 rectangular cross sections, with a spatial integration step equal to SO m. The following values of the model parameters are adopted: p,=2650 kg/m3, ,u=0.20, 4=35", C*=0.75; these values are deduced by field surveys [22].

Figure 3: Areas subjected to flooding with P=0.01 identified by means of FEMA procedure


Figure 4: Areas subjected to floodmg with different values of P=0.01 identified by means of the proposed procedure

The extension of the zones that can be flooded agree with historical observations: the debris flow overtops the levees and the outflowing wave can inundate the town of Fucine. The flow overtops the levees downstream the town of Gianico and a portion of the left riverside can be inundated. Those areas were identified as subjected to flooding by Natale et al. [22] that applied a 2D unsteady model - FL02D [l31 - simulating the propagation of debris flow in a 2D domain, and by Ceriani et al. [20] that applied the Aulitzky method. The areas subjected to flooding shown in Fig. 5 are evaluated by considering the uncertainties in the estimation of the values of the parameters. The limits of three areas outlined in Fig. 5, which refers to P=0.01, are evaluated by considering: the mean values of the parameters (SI, the same of Fig. 4), the uncertainties in the mapping by means of the DPRI method (JP) only, the uncertainties in both the evaluation of the debris flow profiles in Re torrent and the mapping in the alluvial fan (IT).



Figure 5: Areas subjected to flooding for P=0.01 identified by means of the proposed procedure by considering uncertainties in the estimation of the values of the parameters

As described in Section 2.1, triangular probability distribution fimctions are used to describe the variability of the parameters, which are supposed randomly delimited by the limits reported in Tab. 1.

The area of the zones subjected to flooding differs significantly considering and ignoring the uncertainties in the parameters; more specifically, IT area does not reach the main road which intersects the alluvial fan from SW to NE, which can be flooded by considering the mean values of the parameters (area SI). The opposite trend is evaluated for area in the left side of Re torrent downstream Gianico (IT area is the greatest one). These results depend on the geomorphology of the alluvial fan and on the limits of the parameters probability distribution curve and can not be generalised.

Tale 1. Mean values and limits of the parameters Parameter

PS P C*

Mean value 2650 kg/m3

0.20 0.75

Limits 2400-2750 kg/m3

0.10-0.30 0.65-0.85


4 Conclusions and further developments

The proposed method to identify the probability of inundation in alluvial fan is applied to a real world case and the results obtained agree with historical information of past floods. The main advantages in comparison with FEMA procedures concern the possibility of simulating debris flow stages in main channel of the alluvial fan. In t h s way some crucial hydraulic features, such as local changing of the bottom slope, reduction of the wetted area, presence of obstacles (bridges, dams, weirs) are represented and the reaches where the levees are overtopped can be identified and consequently the areas subjected to flooding can be identified in a more accurate way. Moreover the proposed method allow to take into account uncertainties in the definition of the values of the parameters of the model. Although the proposed method requires a systematic application on a wide range of geomorphologic conditions, it seems a useh1 tool for risk assessment. The method must by improved in order to take into account the volume of the debris flow wave and to simulate in a more accurate way the propagation in the alluvial fan by means of 2D models which are already developed by the Authors [23].

Acknowledgments This work is partially hnded by CNR-GNDCI, linea 3.

References

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