# IJEGE-11_BS-Papa-et-alii

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*DOI: 10.4408/IJEGE.2011-03.B-055*

**DERIVATION OF CRITICAL RAINFALL THRESHOLDS FOR DEBRIS**

**FLOW WARNINGS THROUGH MATHEMATICAL AND**

**NUMERICAL MODELING**

**INTRODUCTION**

the morphological, geological and climatic conditions

leading to debris flow formation are quite widespread

and extensive downstream areas result being prone

to debris flow risk. In these cases, the risk reductions

through the building of structural countermeasures

may not only be too expensive but also create en-

vironmental concern. Moreover, in some cases, the

rigid topography of the interested areas, or the lack of

space, makes it difficult to find engineering design and

construction countermeasure solutions.

hazard assessment and civil protection measures are

more suitable in reducing the risks.

on the observations of past events, with the derived

rainfall thresholds therefore depending on the particu-

lar characteristics of the basin from which they have

been derived, and their application on different basins

possibly giving incorrect results. This means that,

theoretically, these approaches may be adopted only

for those basins where a certain amount of observed

debris flow events is available for the derivation of the

threshold line.

**ABSTRACT**

historical events data are not available as well as in the

case of changing environments and climate. For these

reasons, critical rainfall threshold curves are derived

from mathematical and numerical simulations rather

than the classical derivation from empirical rainfall

data. The possible formation of debris flow is simulated

through infinite-slope stability analysis. Land instabil-

ity is governed by the increases of groundwater pres-

sures due to rainfall. The simulations are performed in

a virtual basin, representative of the one studied, taking

into account the uncertainties linked with the defini-

tion of the characteristics of the soil. A large number of

calculations are performed which take into account the

entire range of the governing input dynamic variables

(rainfall characteristics) and different combinations

between them. The dynamic variables considered are

the antecedent rainfall, the intensity of the triggering

rainfall and its duration. The multiple combinations of

the input dynamic variables giving failure is therefore

obtained. For each failure, the corresponding debris

flow volume is estimated. The resulting database is

elaborated in order to obtain rainfall threshold curves.

These curves may be used for the real time evaluation

of possible debris flow events on the basis of observed

and forecasted rainfalls.

**K**

**ey**

**words**

**:**debris flows, warning, critical rainfall thresholds*M.N. PAPA, V. MEDINA & A. BATEMAN*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

phy, the solid concentration, the rheological properties

of the debris mixture and the flow discharge as well as

the occurrence of liquefaction of the sliding mass. In

relation to a specific basin, many of these factors may

be considered as not time dependent. The most rainfall

dependent factors are flow discharge and correlated to-

tal debris volume. In the present study, the total volume

that is instable, and therefore available for the flow,

is considered as the governing factor from which it is

possible to assess whether a debris flow will affect the

downstream areas or not.

the governing input dynamic variables (rainfall char-

acteristics), considering the different possible combi-

nations between them. For any possible combination

of rainfall intensity, duration and antecedent rain, the

total debris volume, available for the flow, is estimat-

ed. The resulting database is elaborated in order to ob-

tain rainfall threshold curves. When operating in real

time, if the observed and forecasted rainfall exceeds a

given threshold, the corresponding probability of de-

bris flow occurrence may be estimated. Warning for

possible debris flow occurrence may be given congru-

ently with such result.

**SIMULATION METHODOLOGY**

*MATHEMATICAL MODEL*

depending on different values of the driving input (rain-

fall characteristics). Consequently the model to assess

possible land instability must be extremely fast. An-

other point is that, in practical cases, the availability of

data about the characteristics of the soil is quite scarce.

hazards may change in response to changing climate,

land use or large forest fires.

a model that reflects the physics of the phenomenon.

Such a system should provide the link between rainfall

and possible debris flow hazards. The system should

be based on simple rules in order to be fast enough to

make possible real time applications.

in which land instability is governed by the increase in

groundwater pressure due to rainfall.

one. The implementation of these models requires an

accurate characterization of the spatial distribution of

the soil properties, which, in many practical applica-

tions, is not available.

conditions (m

i

*et alii*, 2008) have shown that

tion on pore water response and consequent effects on

slope stability improves the effectiveness of regional

shallow landslide hazard maps.

higher than it should be for a wide and safe applica-

tion of the method.

not feasible in the case of warnings due to the long run-

ning time required for this kind of model as well as the

lack of detailed information on the spatial distribution

of the properties of the material in many practical cases.

instable elements along the basin but only if a debris

flow may affect the vulnerable areas in the valley. The

capability of a debris flow of reaching the downstream

*Fig. 1 - Schematic of the slope stability model*

**DERIVATION OF CRITICAL RAINFALL THRESHOLDS FOR DEBRIS FLOW WARNINGS THROUGH MATHEMATICAL AND**

**NUMERICAL MODELING**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*k*

*x*

*is the*

*A*is the drained catchment,

*b*is the width of the slope

element along the direction tangent to the local topo-

graphic contour.

hypothesis of vertical infiltration. Moving from this

assumption, under the hypothesis of wet initial condi-

tions, and with the boundary conditions of transient

groundwater vertical flux equal to zero at great depths

below the water table and water entry at ground sur-

face governed by Darcy’s law, i

*ψ(Z,0)*is the ground water pressure head at the

beginning of the event rainfall,

*I*

*z*

*k*

*z*

tion and

*R(T*)*is defined as follows:

*D*

*0*

soil is near to saturation.

static and dynamic.

limited and the computational time is quite short.

stability analysis (i

*t*), the

factor of safety (

*FS*) is computed by the ratio between

the resisting Coulomb friction and the driving stresses

induced by gravity:

*α*is the slope degree, Z is the vertical coor-

*c*is the soil cohesion,

*φ*is the

angle of internal friction, γ

*ψ*(Z,t) is the ground water pressure head that depends

on depth and time (t).

*FS*is reached (e.g. Fs=1)

*Z*depth is considered instable.

by a short term-heavy rainfall (C

sition of the effect of an “antecedent” rainfall and an

“event” rainfall. The groundwater pressure response to

antecedent rainfall is used as the initial condition for the

time-dependent computation of the groundwater pres-

sure response to the event rainfall.

are reached and the direction of the groundwater flux

may be assumed to be slope parallel. Under this condi-

tion, the ground water pressure may be calculated by:

*d*is the water table depth, measured in the Z

*Z*

*T*

*is the depth of the impermeable bed, (*

*Iz*)

*Tab. 1 - List of static and dynamic input variables*

*M.N. PAPA, V. MEDINA & A. BATEMAN*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

ues of the range of possible values along with the total

number of values for that specific variable,

*m*

*i*

*i*∈

*(1,3).*

static input matrix as well as for each combination of

the dynamic input variables. The total number of simu-

lations (N) is therefore given by:

bris are provided as outputs.

cedent rain (

*I*

*z*

percentage of instable cells or a fixed value of the total

debris volume.

**APPLICATION OF THE MODEL TO A**

**STUDY CASE**

*STUDY AREA*

basin is about 6.3 km

south. The conoid of the basin hosts the ancient vil-

lage of Minori. The outlet of the basin flows into the

Mediterranean Sea.

cient debris flow on the Amalfi Coast, particularly in

1954 and 2005 (C

found between morphotypes, depositional processes

and soil characteristics, with a detailed map of the

soil deposits being subsequently drawn up. The map

was further elaborated, and matched with informa-

tion from literature (b

*et alii*. 2003; I

*et alii*., 2005) in order

parameters (depth of soil layers, geotechnical prop-

erties, hydraulic conductivity, etc.) relevant to the

stability analysis.

*MODEL IMPLEMENTATION*

basin element with given characteristics (static input

variables) subjected to a given rainfall (dynamic input

variables).

lation, previously described, is performed for a certain

amount (n) of computational elements that may rep-

resent the behaviour of the entire basin. The compu-

tational elements do not correspond to the real basin

“pixel” but are virtual elements defined by a string of

the static input variables.

piled with the following procedure. The basin is di-

vided into districts, each one with homogeneous geo-

morphological characteristics and consequently the

same value of the soil variables:

*Z*

*T*

*γs,φ, c, k*

*x*

*k*

*z*

*D*

*0*

close to saturation, may assume any positive value de-

pending on the closeness to complete saturation. On

the basis of this consideration,

*D*

*0*

*/ kz*has been used

variable an average value along with a confidence in-

terval. The entity of the confidence interval is decided

for each variable depending on the methodologies

used for evaluating the variable and the consequent

uncertainties in the evaluation results. The assignment

of a specific soil variable value to a certain number

of input strings follows the normal distribution func-

tion of that variable having the assigned average and

confidence interval

*α*and

*A/b*are comput-

elaborating the Digital Elevation Model of the basin.

Subsequently, the value of

*α*and

*A/b*are assigned to

the strings coherently to the frequency distribution of

the values in the real basin.

computational time and on the other, the representa-

tiveness of the input matrix.

*T, I*

*z*

*, (I*

*z*

**DERIVATION OF CRITICAL RAINFALL THRESHOLDS FOR DEBRIS FLOW WARNINGS THROUGH MATHEMATICAL AND**

**NUMERICAL MODELING**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

was affected. The 1954 event caused a large amount of

damage to the downstream village of Minori. Observa-

tion of the 2005 event showed that about 0.3% of the

basin area was mobilized but the generated debris flow

did not reach the downstream village of Minori. It may

be concluded that the failure percentage of the order of

magnitude of 0.1% does not constitute a hazard for the

village. The threshold value of the failure percentage

should be between 0.3% and 2.8 %.

*RAINFALL THRESHOLDS*

number of computational elements (number of strings

of the static input variables) has been set equal to 1000

torical event observations (see below).

entire range of possible values (Tab. 3).

of the antecedent rain.

average values and the confidence intervals of the soil

variables:

*Z*

*T*

*γ*

*s*

*, φ, c, k*

*x*

*k*

*z*

*.*have been estimated.

*γ*

*s*

*Z*

*T*

*k*

*x*

*k*

*z*

*A/b*are

with a resolution of 5 m.

*Fig. 2 - Map of the homogenous geo-morphological*

*districts (nn stands for absence of soil layer)*

*Tab. 2 - Average values of the soil input variables*

*Fig. 3 - Map of the slope degree*

*M.N. PAPA, V. MEDINA & A. BATEMAN*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

putational elements that result to be instable over the

total amount of elements (failure percentage).

event that occurred in October 1954 (Fig. 4). The

months before that event were dry and therefore the

comparison was carried out with the results obtained

for the antecedent rain equal to zero. The compari-

son shows a very good agreement, with it resulting

through the simulation that for an event rainfall with

a duration in the range of 8-11 hours, the failure per-

centage is about 3%.

(C

*et alii*, 2000) is also reported. In this

as a parameter. From the comparison with the simu-

lation results, it may be concluded that, in the case

of antecedent rainfall equal to zero, the threshold line

proposed by C

*et alii*(2000) corresponds to

sideration made in the previous paragraph, this failure

percentage should not represent a significant hazard

for the downstream village

line proposed by C

*et alii*(2000) corre-

of the totalbasin area. For antecedent rainfall equal

to 120 mm/month (Fig. 6) the same percentage in-

creases up to 1.5%.

for DF warnings. In this kind of graph, the simulation

results are elaborated in order to show, for any ante-

cedent rain, the intensity duration rainfall curve giving

place to a fixed value of the total amount of available

debris volume.

downstream village was 300˙000 m

*et alii*corresponds to

*Tab. 3 - Range of values of the dynamic input variables*

*Fig. 4 - Intensity duration curves for different percenta-*

*ges of instable cells, with antecedent rain equal*

*to zero*

*Fig. 5 - Intensity duration curves for different percenta-*

*ges of instable cells, with antecedent rain equal*

*to 60 mm/month*

*Fig. 6 - Intensity duration curves for different percenta-*

*ges of instable cells, with antecedent rain equal*

*to 120 mm/month*

**DERIVATION OF CRITICAL RAINFALL THRESHOLDS FOR DEBRIS FLOW WARNINGS THROUGH MATHEMATICAL AND**

**NUMERICAL MODELING**

account the liquefaction process.

during the run-out process. It means that a huge vol-

ume of sediment could be produced even if the debris

flow is of a small size in the occurrence area. In these

cases the volume of debris flow calculated by the

method proposed here would be underestimated. On

the other hand an over estimation of the total debris

flow volume may occur because the single computa-

tional element that results to be instable is not really

going to move if it is surrounded by stable elements.

A criterion should be found to fix correctly the criti-

cal percentage of instable cells taking into account

both these problems.

ther validation and calibration of the system is required

of the time interval to be taken into account should be

defined depending on the basin characteristics, while

the loss for evapo-transpiration should be taken into

account in the comparison with past or future events.

parison with observed intense rainfall events that did

not gave place to instability.

**ACKNOWLEDGEMENTS**

gramme through the grant of the Collaborative Project

IMPRINTS (IMproving Preparedness and RIsk maN-

agemenT for flash floods and debriS flow events), Con-

tract FP7-ENV-2008-1-226555.

ing kindly provided some partial results of their ongo-

ing studies on the identification and characterization

of the landslides after the intense rainstorm of October

1954 in the province of Salerno.

**CONCLUDING REMARKS AND FUR-**

**THER DEVELOPPEMENTS**

system is based on critical rainfall thresholds, ob-

tained from a mathematical model through numeri-

cal simulations.

this system may be adopted not only in areas where his-

torical events data are not available but also may take

into account changing environments and climate.

and numerical models, to be used for this derivation,

are quite simple and fast, with it therefore being pos-

sible to apply the method to wide areas.

parison of observed and forecasted rainfalls with the

graphs derived off line.

provements could be made by removing some of the

hypotheses made for the sake of simplicity. A complete

solution of the transient underground water flow may

be implemented instead of the considered simple cases

of slope parallel or vertical flow. In order to assess if a

local instability may develop into a debris flow, a fur-

*Fig.7 - Intensity duration curves for different total debris*

*volume (1000 m*

*3*

*), with antecedent rain equal to 0*

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