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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
273
DOI: 10.4408/IJEGE.2011-03.B-032
STUDY ON THE CONFIGURATIONS OF DEBRIS-FLOW FANS
Y.F. TSAI
(*)
, H.K. TSAI
(**)
& Y.L. CHENG
(***)
(*)
Professor, Department of Social and Regional Development, National Taipei University of Education, Taipei, Taiwan
No.134, Sec. 2, Heping E. Rd., Da-an District, Taipei City 106, Taiwan, tyf@tea.ntue.edu.tw, +886-2-2732-1104 ext. 2236
(**)
Assistant Research Fellow, Institute of Information Science, Academia Sinica, Taipei, Taiwan
28 Academia Road, Section 2, Nankang, Taipei 115, Taiwan, hktsai@iis.sinica.edu.tw, +886-2-27883799 ext. 1718
(***)
Research assistant, Department of Social and Regional Development, National Taipei University of Education, Taipei, Taiwan
No.134, Sec. 2, Heping E. Rd., Da-an District, Taipei City 106, Taiwan, superdaiwa@gmail.com, +886-2-2732-1104 ext. 2314
scribed as gravity flows of a mixture of soil, rocks, water,
and/or air. As a landslide-initiated debris flow proceeds
down a steep mountain canyon, it scours materials from
the channel and grows in mass. When the debris flow
reaches a gentle basin from the steep channel, it spreads
out, reduces its momentum and then stops after reaching
a flatter area. Sediment deposits, leaves mud fluid, or
clear water flowing downstream. This process gradually
creates a debris-flow fan. The fan of debris-flow usually
becomes a hazard zone and makes a serious damage and
property loss. Therefore, the study on the hazard zone
mapping of debris-flow become very important.
The process of formation and the shape of debris-
flow fans have received attention by investigators
such as H
ooke
(1967), o
kuda
(1973), o
kuda
et alii
(1977), t
akaHasHi
& y
osHida
(1979), t
akaHasHi
(1980), and a
sHida
et alii (1980). These investigations
provided information on local debris-flow fans, but
did not consider the morphological similarity. Numer-
ous researchers tried to identify the hazard zone of de-
bris flow based on the maximum deposition length of
debris-flow fan. t
akaHasHi
(1991) combine the law of
conservation of mass with momentum balance equa-
tion to derive the deposition length of debris flow fan.
o
kuda
(1973) utilized the particle dynamics method
to estimate the length of debris-flow fan. C
annon
(1993), o
kuda
(1984), i
keya
(1981) and b
atHuRst
et alii (1997) make use of statistical analysis based
on historical data of field investigation to predict the
spread length of debris flow. Although these previous
ABSTRACT
Studies on debris-flow fan configurations are fun-
damental to map hazard zone of debris flow disasters.
This study aims to identify the morphological similarity
of debris-flow fans based on a series of laboratory ex-
periments and field investigations. The maximum length
Lc, width Bm and thickness Zo of debris-flow fans are
adopted as the characteristic parameters for analyzing
the morphological similarity. The analysis demonstrates
that the non-dimensional longitudinal and transverse
profiles of debris-flow fans are described by Gaussian
curves, while the non-dimensional plan form is closely
fitted using a circular curve. By combining the three
non-dimensional curves, the three-dimensional topog-
raphy of debris-flow fans are easily obtained based on
the parameters L
c
, B
m
and Z
o
. The volume V are simply
related to these maximum values via V=αL
c
B
m
Z
o
with
an empirically determined shape parameter α. From the
result of verification with a debris flow fan in Foncho
Village, Nan-tou County of central Taiwan, it shows that
the 3D topography function makes a good hazard zone
mapping of debris flow. Furthermore, the parameter α is
approximately 0.28 for a natural stony debris-flow fan.
K
ey
words
: debris flow, debris-flow fan, morphological si-
milarity
INTRODUCTION
Debris flows are highly hazardous hydrological
processes common in Taiwan. They are generally de-
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Y.F. TSAI, H.k. TSAI & Y.L. CHENG
274
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
flood basin, respectively; Q
d
=discharge of debris flow
supplied at the upstream end of the main flume, and
it is equal to the dry sediment discharge plus the mud
flow or pure water discharge; C
d
=the sediment volume
concentration of debris flow formed in the main flume
(computed as the ratio of dry sediment discharge and
debris flow discharge); ρ=density of supplied pure wa-
ter or mud fluid; L
c
, B
m
, and Z
o
denote, respectively,
the measured maximum values of length, width, and
thickness of debris-flow fans; and V is the measured
volume in the flood basin for each experiment.
Notably, three kinds of sediments with uniform
size (C
u
=1.0) were used in experiments of Runs 1-30
(d
50
=1.6, 4.5, and 7.0 mm) to investigate the effects
of sediment size, sediment volume concentrations,
incline angles of the main flume, and the flood basin
on the shape of a stony debris-flow fan. Sediments
with granular mixture size (C
u
=2.5 to 13) were used
in the experiments conducted in Runs 31-57 to exam-
ine the influence of size distribution on the shape of
a stony debris-flow fan. On the other hand, sediments
with granular mixture size (C
u
=1.0 to 13) and mud
fluid (ρ=1.0 to 1.365) were used in the experiments of
Runs 58-72 to study the effect on the shape of a mud
debris-flow fan. The mud-fluid mixes lean clay and
pure water with different volume concentrations (C
v
)
of 5, 13, and 22%. The median diameter of lean clay
is about 0.016 mm. The densities (ρ) of the three mud
fluids used in the experiments were 1.083, 1.216, and
1.365 g/cm
3
. Similar experiments were conducted
by o’b
Rien
& J
ulien
(1986) and m
aJoR
& P
ieRson
(1992). Table 1 also lists three experiments (name-
ly, Runs T1, T2, and T3) performed by t
akaHasHi
(1980) to provide a comparison.
studies can approximately map the hazard zone they
did not consider the three dimensional (3D) topogra-
phy of the debris-flow fans.
Many researchers, such as s
HieH
et alii (1996),
R
iCkenmann
& k
oCH
(1997), f
RaCCaRollo
& P
aPa
(2000), l
aiGle
et alii (2003), C
etina
& k
Rzyk
(2003),
and i
toH
et alii (2003) developed numerical models of
debris-flow depositions on fans and made good simu-
lations. While these numerical models can offer more
precise simulations of debris-flow fans, but complex cal-
culation conditions are required and these models some-
times need to adjust the models’ parameters case by case.
This study aims to identify the hazard zone of
debris flow via a series of investigation on the debris-
flow fan configurations. We find that a simple relation
exists between the volume of a debris-flow fan and its
maximum length, width, and thickness from a series
of experiments. The 3D topography of a debris-flow
fan is also derived from this study. By utilizing the
3D topography of a debris-flow fan, we can easily
map the hazard zone of debris flow. For the purpose
of verification, a debris-flow fan in Foncho Village,
Nantou County was investigated. The results reveal
that the morphological deposition method can map the
approximate hazard zone of debris flow.
LABORATORY EXPERIMENTS
EXPERIMENTAL SETUP AND CONDITIONS
A series of experiments was performed in labo-
ratory to observe the formation process of a debris-
flow fan at a steep channel mouth. The experimental
facility includes a main flume and a flood basin. Fig.
1 schematically shows the experimental set-up. The
main flume is 8.0-m long, 0.2-m wide, and 0.5-m
deep. The flood basin with transparent sidewalls is
closely connected with the downstream end of the
main flume and is 3.0-m long, 2.2-m wide, and 0.5-m
deep. The main flume can be tilted from 0to 30°, and
the flood basin can be tilted from 0 to 20°.
Seventy-two tests were conducted with varying
gradients of the main flume and flood basin, sediment
size and size distribution, supplied water, mud-fluid
and sediment discharges. Table 1 summarizes the con-
ditions of each experiment. In Table 1, d
50
=mean di-
ameter of the used sediment materials; C
u
=uniformity
coefficient defined as d
84
/d
16
, in which d
16
(or
d
84
)=sediment size for which 16 (or 84)% of the sample
is finer; θ
1
and θ
2
=gradients of the main flume and the
Fig. 1 - Schematic diagram of experimental setup in top
view and side view
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STUDY ON THE CONFIGURATIONS OF DEBRIS-FLOW FANS
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275
Tab. 1 - Experimental conditions and results in the study
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Y.F. TSAI, H.k. TSAI & Y.L. CHENG
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
some distance from it. Consequently, the debris-flow
fan in the flood basin maintained its own shape and
remained unchanged for a long period. This study re-
fers to this stage of debris-flow fan development as the
“quasi-stable stage.” During this stage, the deposition
process proceeded upstream along the main flume,
and the great part of the sediments was deposited
on the main flume. Based on experimental observa-
tions, there are two important characteristics when the
debris-flow fan reached the quasi-stable stage. First,
there is a long-term subsidence of the debris-flow fan
in the flood basin. Second, the maximum width and
maximum thickness of the debris-flow fan will locate
at the outlet of the main flume. On the other hand, if
low-concentrated flow or water flow follows debris
flow, the debris-flow fan will be eroded.
Fig. 4 displays the definitions of the debris-flow
fan characteristic parameters for Run 10. Specifically,
Fig. 4(a) shows the top view of the fringed shape of
the debris-flow. At the intersection of the main flume
and the flood basin, X=0. The positive value of X is
downstream of the debris-flow fan. The central line of
the debris-flow fan is indicated with Y=0, with positive
values of Y denoting the left side of the debris-flow fan
(viewed from upstream to downstream). Moreover, B
is the width of the debris-flow fan at any section, and B
(X) decreases with increasing X. The maximum width
of the debris-flow fan is B
m
, and usually occurs at X=0
in the quasi-stable stage. Fig. 4(b) shows the longitu-
dinal profile of the debris-flow fan. The length of the
debris-flow fan is measured from the apex along the
longitudinal axis. Usually, the maximum length is de-
PROCESS OF FORMATION OF DEBRIS-FLOw FAN
The experimental results of Run 53 were adopted
to describe the formation process of the debris-flow
fan. The slopes of the main flume and flood basin for
the case of Run 53 were set to 19° and 2°, respective-
ly. Fig. 2 shows temporal variations in the shapes of
Run 53. The time, indicated in the figures, was count-
ed from the moment the forefront of the debris flow
arrived at the mouth of the main flume (namely, the
entrance of the flood basin). Fig. 3 shows the devel-
opment of the maximum values of deposition length,
width, and thickness of Run 53. In the earliest stage,
when the debris flow developed in the main flume
debouched onto the flood basin, the path of the flow
downstream from the main-flume outlet was straight,
and the deposition length of the debris-flow fan
quickly reached its maximum at 6 s. For a while, the
flow drifted rightward and leftward, and its width and
thickness build up with time. The deposition width
of the debris-flow fan reached its maximum at 22 s,
and the deposition is located on the connection be-
tween the main flume and flood basin. The deposition
thickness of the debris-flow fan reached its maximum
at 25 s and is located at the outlet of the main flume
(namely, the apex of debris-flow fan). The period of
fan development before 25 s is called the “developing
stage.” During this stage, the maximum thickness and
maximum width are not located at the outlet but at
Fig. 2 - Temporal variations in shapes of Run 53
Fig. 3 - Developments of the maximum values of deposi-
tion length,width, and thickness of Run 53
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STUDY ON THE CONFIGURATIONS OF DEBRIS-FLOW FANS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
277
noted by L
c
. The deposition thickness at X=0 and Y=0
is represented by Z
o
, which is usually the maximum
thickness of the debris-flow fan in the quasi-stable
stage. Fig. 4(c) illustrates the transverse profile, where
Z denotes the deposition thickness at any point, and Z
c
represents the deposition thickness at the central point
along the longitudinal axis. The variation of the debris-
flow fan beyond the quasi-stable stage is not studied in
this paper. This investigation focuses on the shape of
the debris-flow fan in the quasi-stable stage.
DEBRIS-FLOW FAN CONFIGURATIONS
MORPHOLOGICAL SIMILARITY OF TRAN-
SVERSE PROFILE
Thirty runs (Runs 1-30) of the present experimen-
tal results using uniform sediments (Cu=1.0), as well
as three runs of Takahashi’s (1980) (Runs T1-T3) were
adopted to analyze the transverse profile of stony de-
bris-flow fan in the quasi-stable stage. About 280 trans-
verse profiles were surveyed to analyze from Runs 1-30
and Runs T1-T3. The non-dimensional values of Z/Z
c
against Y/B are illustrated in Fig. 5 and their relation-
ship was approximated by a Gaussian curve with an
empirical coefficient C
T
, represented as Eq. 1.
The best-fitting coefficient C
T
=0.211, and the
correlation coefficient is around 87%. In these experi-
ments, the range of C
T
is around 0.204~0.225, which
varies insignificantly. A smaller C
T
means a steeper
Gaussian distribution. For stony debris-flow experi-
ments (Runs 31-57) and mud debris-flow experiments
(Runs 58-72), the non-dimensional transverse profile
could also be approximated using a Gaussian curve,
but the C
T
increases with the uniformity coefficient
C
u
and the mud-fluid density ρ. For a natural stony
debris-flow fan (with a very large C
u
and ρ=1.0), the
value of C
T
is approximately 0.248.
MORPHOLOGICAL SIMILARITY OF LONGITU-
DINAL PROFILE
Thirty runs (Runs 1-30) of the present experimen-
tal results with uniform sediments (Cu=1.0) and three
runs of Takahashi’s (1980) (Runs T1-T3) were used to
analyze the longitudinal profile of stony debris-flow
fan in the quasi-stable stage. Fig. 6 plots the non-di-
mensional values of Z
c
/Z
o
against X/L
c
, and their rela-
tionship were approximated by a half-Gaussian curve
with an empirical coefficient C
L
, represented as Eq. 2.
Fig. 4 - Definitions of debris-flow fan characteristic
parameters
Fig. 5 - Relationship between Z/Zc and Y /B via a Gaus-
sian curve
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Y.F. TSAI, H.k. TSAI & Y.L. CHENG
278
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
By the Eq. 4, the volume formula of debris-flow
fan was derived from Z =
f (X, Y, L
c
, B
m
, Z
o
), repre-
sented as Eq. 5
V = α L
c
B
m
Z
o
Where α=function of the shape parameters C
T
and
C
L
, represented as Eq. 6.
The best Gaussian distribution coefficients,
C
T
and C
L
, for each run were substituted into Eq.
6 to yield the values α. For 30 runs (Runs 1-30)
of the present experimental results using uniform
sediments (Cu=1.0) as well as three runs of t
aka
-
HasHi
s
(1980) (Runs T1-T3), C
T
=0.211, C
L
=0.393
and α is about 0.24 from Eq. 6. For a natural stony
debris-flow fan (with a very large C
u
and ρ=1.0),
C
T
=0.248, C
L
=0.416, and α is approximately 0.275
from Eq. 6. By Eq. 5 V=αL
c
B
m
Z
o
, via an empirical
coefficient α, the volume V of debris-flow fan is
related to the maximum length L
c
, width B
m
, and
thickness Z
o
. Therefore, the 3D topography of a
debris-flow fan is easily derived based on the pa-
rameters L
c
, B
m
, and Z
o
using three morphologi-
cally similar formulas. Namely, after L
c
, B
m
, and Z
o
is estimated, and the potential hazard zone of each
debris-flow field is obtained.
The best-fitting coefficient C
L
=0.389, and the cor-
relation coefficient is approximately 91%. In these ex-
perimental runs, the range of C
L
is around 0.373~0.401.
This variation is insignificant. For stony debris-flow
experiments (Runs 31-57) and mud debris-flow experi-
ments (Runs 58-72), the non-dimensional longitudinal
profile were also approximated via a half-Gaussian
curve. However, the results shows that C
L
increases
with the uniformity coefficient C
u
and fluid density ρ.
For a natural stony debris-flow fan (with a very large
C
u
and ρ=1.0), the value of C
L
is approximately 0.416.
MORPHOLOGICAL SIMILARITY OF PLANE
PROFILE
Let B denote the width of a stable debris-flow fan
at any section X, while B
m
represents the maximum
width of that flow fan, which generally occurs at X=0;
L
c
=maximum deposition length. Fig. 7 illustrates the
relative width B/B
m
as a function of the relative coordi-
nate X/L
c
. The relationship between B/B
m
and X/L
c
of 75
runs (Runs 1-72 and Runs T1-T3) were approximated by
a quarter circle curve, with a correlation coefficient of up
to 98%. The circle function represented as Eq. 3.
THREE-DIMENSIONAL TOPOGRAPHY OF
DEBRIS-FLOw FANS
The 3D topography of debris-flow fan was de-
rived as Eq. 4, obtaining from Eq. 1, 2, and 3.
Fig. 6 - Relationship between Z
c
/Z
o
and X/L
C
, via a half-
Gaussian curve
Fig. 7 - Relation between B/B
m
and X/L
c
via a quarter
circle curve
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STUDY ON THE CONFIGURATIONS OF DEBRIS-FLOW FANS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
279
According to the equilibrium concentration for-
mula of debris flow derived by t
akaHasHi
(1977),
represented as Eq. 7. The volumetric concentration of
debris flow has an upper limit, where the value C
d
is
about 0.9 times of C
*
.
2. The estimate of flow discharge (Q
d
) (cms) of de-
bris flow;
The relation between Q
d
and rainfall intensity (I)
(mm/hr) is shown as Eq. 8. Where C represents the run-
off coefficient (assume as 0.8). The rainfall intensity (I)
is used with 25-year frequency (I
25
= 132.1 mm/hr).
3. The estimate of mean velocity (u) (m/sec) of de-
bris flow;
With the constitutive relation b
aGnold
(1954) de-
rived to predict the mean velocity (u) of debris flow,
the Equation is shown as Eq.9. Where α
i
is friction
coefficient (=0.073), g is acceleration of gravity.
4. The estimate of mean flow depth (h) (m) of debris flow;
According to the law of conservation of mass in
fluid mechanics, Q
d
is represented as Eq. 10. The mean
flow depth (h) of debris flow was derived from Substi-
tuting Eq.8 and Eq. 9 into Eq. 10 which yields Q
d
= f (h)
Q
d
= B
0
hu
After this series of calculations, the hydrological
characteristics of debris flow located in Foncho Vil-
lage during the period of Typhoon Herb could be ob-
tained respectively. The value of flow discharge (Q
d
),
mean velocity (u), mean flow depth (h) and volumet-
ric concentration (C
d
) is approximately 493.2cms,
18.1m/sec, 2.27m and 58.5%.
THE CALCULATION OF THREE-DIMENSIO-
NAL TOPOGRAPHY OF DEBRIS-FLOw
By combining the volume formula of debris-flow
fan derived from the experiments and hydrological
data, the 3D topography of a debris-flow fan can be
easily derived based on the parameters L
c
, B
m
, and Z
o
.
For this purpose, various theory and empirical formu-
las were used to calculate the debris-flow hazard zone
of Foncho Village in Nantou county.
1. The estimate of maximum length (L
c
) of debris-
flow fan.;
CASE STUDY
STUDY AREA
The debris flow fan located in Foncho Village,
Nantou County, Taiwan were studied in this study. We
derived the basic data of the stream. The watershed
area (A) of the stream is about 3 ha; length (L) of the
stream is about 3 km; mean slope (θ
1
) of the main flume
is about 28°; mean slope (θ
2
) of the flood basin is 8.7°.
Debris-flow disaster was common in this area. The
most serious disaster had occurred during the period of
Typhoon Herb in1996. Fig. 8 shows the debris-flow fan
which occurred after Typhoon Herb. In this event, the
discharged volume of debris flow was up to 450000cm
3
.
The characteristic of sediments of debris-flow fan was
investigated. Where B
o
denotes the mean slope of up-
stream is about 12m; mean diameter (d
50
) of the materi-
als is about 3.5cm; uniformity coefficient (C
u
) is about
67.5; internal friction angle (φ) is about 35°; kinematic
friction angle (φ
k
) is about 29°; density (ρ) is about 2.65
g/cm
3
; sediment volumetric concentration (C
*
) is about
0.65. In addition, the quantity of fine materials (median
diameters < 0.1mm) is under 10%. With the characteris-
tics mentioned above, the debris flow in Foncho Village
was treated as a stony debris flow.
In order to understand the relation between the
debris-flow hazard zone and hydrology, various theo-
ries and empirical formulas were used to calculate the
hydrological characteristics of debris flow during the
period of Typhoon Herb.
1. The estimate of volumetric concentration (C
d
) of
debris flow;
Fig. 8 - Debris-flow fan located on Fon-
cho Village in Central Taiwan
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Y.F. TSAI, H.k. TSAI & Y.L. CHENG
280
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
In order to obtain the maximum length (L
c
) of de-
bris-flow fan, a formula t
akaHasHi
(1991) proposed
was adopted. Based on conservation of mass and mo-
mentum balance, the Equation was revised as Eq. 11:
where:
K
a
means the coefficient of active earth pressure,
in which K
a
=tan
2
(45°-φ/2 ).
2. The estimate of maximum deposition thickness
(Z
o
) of debris-flow fan;
From the Fig. 4(b), the relation between maximum
deposition es (at the apex of fan), mean deposition
slope along the longitudinal axis (r) and maximum
length (L
c
) is represented as Eq. 12. Namely, if the val-
ue of r and L
c
could be obtained, the maximum depo-
sition thickness (Z
o
) could be estimated from Eq. 12.
The deposition slope of debris-flow fan was de-
rived from t
akaHasHi
(1991). Eq. 13 represents the
critical deposition slop of debris-flow fan, where the
H
o
represents a flow depth above the deposition layer,
d is diameter of sediment.
The values, H
o
and d were difficult to and field
investigations. By a series of analyses, t
sai
(1999)
had found a close relation between mean deposition
slope along the longitudinal axis (r), unit flow dis-
charge (q), volumetric concentration (C
d
) and sedi-
ment diameters (d
50
). Consequently, the Eq.14 was
derived from Eq. 13.
3. The estimate of maximum width (B
m
) of debris-
flow fan;
If the maximum length (L
c
), mean deposition
slope (r), and the volume of debris fan could be ob-
tained, the B
m
is estimated by Eq. 15. Subtituting Eq.
12 into Eq. 5 yields the Eq.15:
The maximum length, width and thickness of de-
bris flow were estimated by substiuting the hydrologi-
cal characteristics into the formulas mention above.
The maximum length (L
c
) is about 262.4m, the mean
deposition slope (r) is about 11.03°, maximum deposi-
tion thickness (Z
o
) is about 10.6m, and the maximum
width (B
m
) is about 585.5m. By substituting the val-
ues of L
c
, r, Z
o
and B
m
into Eq.4, the 3D topography
function of the debris-flow fan in Foncho Village was
estimated by Eq. 16. Finally the hazard zone mapping
of debris flow could be drawn as shown in Fig. 9.
From the research results compared with the field
investigations (Fig. 9), it shows that the proposed
methods make a good hazard zone mapping of debris
flow. In addition, the longitudinal profiles of central
axis of debris-flow fan ware both very similar in ex-
periments and field investigations, as shown in Fig. 10.
From the Fig. 10, we discovered that the observed new
bed was higher than the simulated new bed only at the
end of the debris-flow fan (at the distance of 200m).
Consequently, the methods this research proposed can
be applied to map the hazard zone of debris-flow fan.
CONCLUSIONS
This study presents labo ents and case study on
the configurations of debris-flow fan. Seventy-five ex-
perimental tests (Runs 1-72, Runs T1-T3) show that
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(13)
(14)
(15)
(16)
Fig. 9 - Topographies of the debris-flow fan in Foncho
Village from experiments and field investigations
Fig. 10 - Changes of the bed of fan between experiments
and field investigations
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STUDY ON THE CONFIGURATIONS OF DEBRIS-FLOW FANS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
281
Village, Nantou County, Taiwan was investigated as
an example. From the experimental results compared
with the case study, it shows that the proposed meth-
ods make a good hazard zone mapping of debris flow.
In conclusion, if L
c
, B
m
, and Z
o
were estimated, the
potential hazard zone of each debris-flow field could
be divided into several different danger zones via the
designed deposition thickness calculated from the 3D
topography function. Such division is very useful for
hazard zone mapping of debris flow.
the profiles of debris-flow fan were approximated by
Gaussian curves in both the longitudinal and trans-
verse directions of the debris-flow path, while the
shape of the fringe resembled a quarter circle curve. By
combining the three non-dimensional curves, the vol-
ume V of debris-flow fan are related to the maximum
length L
c
, width B
m
, and thickness Z
o
by V=αL
c
B
m
Z
o
,
via an empirical coefficient α. The 3D topography of
a debris-flow fan are easily derived based on the pa-
rameters L
c
, B
m
, and Z
o
using three morphologically
similar formulas. Then, a debris-flow fan in Foncho
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Y.F. TSAI, H.k. TSAI & Y.L. CHENG
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