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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
495
DOI: 10.4408/IJEGE.2011-03.B-055
DERIVATION OF CRITICAL RAINFALL THRESHOLDS FOR DEBRIS
FLOW WARNINGS THROUGH MATHEMATICAL AND
NUMERICAL MODELING
m
aRia
n
iColina
PAPA
(*)
, v
iCente
MEDINA
(**)
& a
llen
BATEMAN
(**)
(*)
University of Salerno, Dept. of Civil Engineering - Via Ponte don Melillo - 84084 Fisciano (SA), Italy
E-mail: mnpapa@unisa.it
(**)
Technical University of Catalonia (UPC), Sediment Transport Research Group, Hydraulic, Marine and
Environmental Engineering Department, Spain
INTRODUCTION
Debris and mud flow events may affect vulner-
able areas causing a major natural risk. In some areas,
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.
For these reasons in many cases, non structural
countermeasures, such as warnings through real time
hazard assessment and civil protection measures are
more suitable in reducing the risks.
The most common approach adopted in literature
for real time debris flow hazard assessment (w
ieC
-
zoRek
& G
lade
, 2005) is based on empirically de-
rived rainfall thresholds. These approaches are based
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
The aim of the work is to develop a system capa-
ble of providing debris flow warnings in areas where
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
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M.N. PAPA, V. MEDINA & A. BATEMAN
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
areas depends on many factors linked with the topogra-
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 approach presented is based on the simulation
of a large number of cases covering the entire range of
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
The proposed approach is based on the simulation
of a large number of possible debris flow occurrences,
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.
For these reasons, the selected simulation method-
Another drawback of empirically derived thresh-
olds is that they cannot anticipate how debris flow
hazards may change in response to changing climate,
land use or large forest fires.
In order to overcome these limitations, it is neces-
sary to obtain a debris flow warning system through
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.
Theoretical models of rainfall triggered debris
flows are based on the infinite-slope stability analysis
in which land instability is governed by the increase in
groundwater pressure due to rainfall.
These models are usually implemented in discrete
landscape cells and give the security factor for each
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.
Many of the approaches proposed in literature are
based on the hypothesis of steady groundwater flow
conditions (m
ontGomeRy
& d
ietRiCH
, 1994).
A simplified model that accounts for transient
groundwater flow conditions has been proposed by
i
veRson
(2000).
Comparisons with observed scares of debris flow
formation areas (G
odt
et alii, 2008) have shown that
accounting for the transient effects of rainfall infiltra-
tion on pore water response and consequent effects on
slope stability improves the effectiveness of regional
shallow landslide hazard maps.
However, the number of false positives and false
negatives in the predicted unstable cells is still far
higher than it should be for a wide and safe applica-
tion of the method.
The implementation of a distributed model, based
on the stability analysis for each grid cell of the basin, is
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.
Moreover, with the aim of giving debris flow warn-
ings, it is not necessary to know the distribution of
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
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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
497
slope normal direction, in steady conditions, k
x
is the
hydraulic conductivity in the slope parallel direction,
A is the drained catchment, b is the width of the slope
element along the direction tangent to the local topo-
graphic contour.
Following the approach of i
veRson
(2000), the
short term response to rainfall may be assessed in the
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
veRson
(2000) pro-
posed an analytical solution of the Richards equation
where ψ(Z,0) is the ground water pressure head at the
beginning of the event rainfall, I
z
is the infiltration rate
at ground surface, in the slope normal direction, k
z
is
the hydraulic conductivity in the slope normal direc-
tion and R(T*) is defined as follows:
in which:
where D
0
is the maximum characteristic diffusivity,
governing the transmission of pressure heads when the
soil is near to saturation.
The input variables (Tab. 1) that feed the model
presented above are divided into two main families,
static and dynamic.
ology is extremely simplified, thus the requested data is
limited and the computational time is quite short.
The possible triggering debris flow is simulated,
in a generic element of the basin, by an infinite slope
stability analysis (i
veRson
, 2000; t
ayloR
, 1948). At
any depth from the surface (Z), and at any time (t), the
factor of safety (FS) is computed by the ratio between
the resisting Coulomb friction and the driving stresses
induced by gravity:
where α is the slope degree, Z is the vertical coor-
dinate, positive downward, c is the soil cohesion,φis the
angle of internal friction, γ
s
is the depth averaged soil
unit weight, γ
w
is the unit weight of ground water and
ψ(Z,t) is the ground water pressure head that depends
on depth and time (t).
When a critical value of FS is reached (e.g. Fs=1)
the soil over the Z depth is considered instable.
Many observed debris flow events have been trig-
gered by a long term and low intensity rainfall followed
by a short term-heavy rainfall (C
RozieR
, 1989; w
ieC
-
zoRek
& G
lade
, 2005). As a consequence, the trigger-
ing groundwater pressure is calculated by the superpo-
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.
If the antecedent rainfall has a sufficiently low in-
tensity and long duration, the steady state conditions
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:
where d is the water table depth, measured in the Z
direction, in steady state conditions. Following m
ont
-
GomeRy
& d
ietRiCH
(1994), the mass conservation
equation of groundwater gives the following:
where Z
T
is the depth of the impermeable bed, (Iz)
steady is the infiltration rate at ground surface, in the
(1)
(2)
(3)
(4)
(5)
(6)
Tab. 1 - List of static and dynamic input variables
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M.N. PAPA, V. MEDINA & A. BATEMAN
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
assigned by the definition of the lower and upper val-
ues of the range of possible values along with the total
number of values for that specific variable, m
i
i(1,3).
The simulation is then performed for each row of the
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:
For each simulation, the number of instable ele-
ments and the corresponding volume of available de-
bris are provided as outputs.
The obtained data are then elaborated, by simple
interpolation, in order to build, for each value of ante-
cedent rain (I
z
)
steady,
a graph representing the intensity-
duration rainfall curves producing a fixed value of the
percentage of instable cells or a fixed value of the total
debris volume.
APPLICATION OF THE MODEL TO A
STUDY CASE
STUDY AREA
The area under study is a basin on the Amalfi
Coast (Sambuco Basin), in the south of Italy. The
basin is about 6.3 km
2
, delimited by mountains of
about 800-1000 m a.s.l, heading prevalently north-
south. The conoid of the basin hosts the ancient vil-
lage of Minori. The outlet of the basin flows into the
Mediterranean Sea.
Historical debris flow events are documented in
1910, 1924 and 1954 (P
aPa
& t
Rentini
, 2010).
Morphological and geo-pedological studies
have been conducted on the triggering areas of an-
cient debris flow on the Amalfi Coast, particularly in
1954 and 2005 (C
aRbone
, i
amaRino
& G
allo
, paper
in preparation). From these studies, matches were
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
asile
et alii. 2003; I
amaRino
& t
eRRibile
, 2008; b
ilotta
et alii., 2005) in order
to obtain the geographical distribution of the soil
parameters (depth of soil layers, geotechnical prop-
erties, hydraulic conductivity, etc.) relevant to the
stability analysis.
The basin has been divided into 21 homogeneous
MODEL IMPLEMENTATION
The mathematical model described in the pre-
vious paragraph can assess possible instability of a
basin element with given characteristics (static input
variables) subjected to a given rainfall (dynamic input
variables).
In order to asses if a specific basin may give place
to the formation of a debris flow, the instability simu-
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.
The input of the model is therefore a matrix of
static variables, representing the studied basin, com-
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
and k
z
.
The maximum characteristic diffusivity D
0
governing
the transmission of pressure heads, when the soil is
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
as a calibration parameter.
The uncertainties in the evaluation of the soil
variables are taken into account assigning to each
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
The topographic variables α and A/b are comput-
ed for each basin pixel, with simple GIS instruments
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.
The number of matrix strings (n) may be set as
an input parameter, thus regulating on one hand the
computational time and on the other, the representa-
tiveness of the input matrix.
The dynamic input variables T, I
z
, (I
z
)steady are
(7)
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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
499
From the reconstruction of the areas that were mo-
bilized in 1954 (C
aRbone
, i
amaRino
& G
allo
, paper in
preparation), it resulted that 2.8% of the total basin area
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
The static input variables matrix has been compiled
using the data presented in the previous paragraph. The
number of computational elements (number of strings
of the static input variables) has been set equal to 1000
The calibration parameter D
0
/K
z
has been adjusted
to 0.1 by comparison of simulated results with the his-
torical event observations (see below).
The dynamic input variables have been set with
reference to the local climatology in order to cover the
entire range of possible values (Tab. 3).
The total number of simulations (N) resulted to be
25×10
6
.
The obtained data set has been elaborated in order
to draw intensity-durations curves for any fixed value
of the antecedent rain.
Any intensity duration rainfall curve corresponds
geomorphological districts (Fig. 2), for each district the
average values and the confidence intervals of the soil
variables: Z
T
γ
s
, φ, c, k
x
and k
z
. have been estimated.
The average values of the soil variables are report-
ed in Tab. 2. The confidence intervals were set, for γ
s
and j equal to the 10% of the average while for the Z
T
,
c, k
x
and k
z
equal to the 20% of the average.
The topographic variables a (Fig. 3) and A/b are
calculated by elaborating a Digital Elevation Model
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
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M.N. PAPA, V. MEDINA & A. BATEMAN
500
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
to a fixed value of the ratio between the amount of com-
putational elements that result to be instable over the
total amount of elements (failure percentage).
The simulation results are compared with the
rainfall curve relative to one observed debris flow
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%.
In the same graph of Fig. 4, a rainfall threshold
line, derived through the elaboration of empirical data
(C
alCateRRa
et alii, 2000) is also reported. In this
case, the antecedent rainfall does not explicitly appear
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
alCateRRa
et alii (2000) corresponds to
a failure percentage of about 0.1%. Following the con-
sideration made in the previous paragraph, this failure
percentage should not represent a significant hazard
for the downstream village
If the same comparison is made with the simula-
tion results obtained with antecedent rainfall equal
to 60 mm/month (Fig. 5), it results that the threshold
line proposed by C
alCateRRa
et alii (2000) corre-
sponds to a percentage of instable area equal to 1%
of the totalbasin area. For antecedent rainfall equal
to 120 mm/month (Fig. 6) the same percentage in-
creases up to 1.5%.
Once the volume threshold is fixed, a graph simi-
lar to the one shown in (Fig. 7) may be used as a rule
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.
In the example of the DF in the Sambuco basin
in 1954, the total debris flow volume that reached the
downstream village was 300˙000 m
3
. The threshold
line proposed by C
alCateRRa
et alii corresponds to
about 5˙000 m
3
.
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
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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
501
ther condition should be added to the model, taking into
account the liquefaction process.
In many cases, a great increment of debris flow
volume occurs as a consequence of channel erosion
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.
The system has been tested through comparison
with only one test basin and one debris flow event. Fur-
ther validation and calibration of the system is required
A more detailed study should be carried out, deal-
ing with the antecedent rainfall. The reference length
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.
An assessment of the “false alarm” that may be
given by the system can be carried out through the com-
parison with observed intense rainfall events that did
not gave place to instability.
ACKNOWLEDGEMENTS
The work described in this study was supported
by the European Community Seventh Framework Pro-
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.
The authors would like to thank the dr Antonio
Carbone, Antonio Gallo and Michela Iamarino for hav-
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
In the present study, a simple system was devel-
oped, to be used as a debris flow warning tool. The
system is based on critical rainfall thresholds, ob-
tained from a mathematical model through numeri-
cal simulations.
In contrast to widespread critical rainfall thresholds
derived from past debris flow events measurements,
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.
The derivation of the critical rainfall thresholds
curves has to be performed off line. The mathematical
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.
A set of rainfall threshold curves is provided, one
for each considered value of antecedent rainfall.
In real time, it is possible to provide replies in
very short time intervals through the simple com-
parison of observed and forecasted rainfalls with the
graphs derived off line.
The model used for describing the mechanism of
debris flow formation is very simple and many im-
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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ele
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