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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
941
DOI: 10.4408/IJEGE.2011-03.B-102
PROBABILITY DISTRIBUTION OF DEBRIS FLOW DEPTH
AND ITS IMPLICATION IN RISK ASSESSMENT
y
onG
li
(*, **)
, P
enGCHenG
su
(*, **, ***)
& J
inGJinG
liu
(*, **, ***)
(*)
Key Laboratory of Mountain Hazards and Surface Process, Chinese Academy of Sciences, Chengdu, 610041, China
(**)
Institute of Mountain Hazards and Environment, Chinese Academy of Sciences & Ministry of Water Conservancy
Chengdu, 610041, China
(***)
Graduate School of Chinese Academy of Sciences,Beijing,100039, China
C
osta
& J
aRRett
, 1981; s
oHn
, 2000; s
HaRP
& n
obles
,
1953; J
oHnson
, 1970; C
osta
, 1984; w
HiPPle
& d
unne
,
1992; d
e
G
Raff
, 1994). As the living surge is hardly
seen in the field, the deposit proves the “fossil” for
deriving parameters of debris flow as fluid of visco-
plasticity (m
iddleton
& H
amPton
, 1973, 1976; l
owe
,
1975, 1976, 1982; C
oussot
et alii, 1996).
Deposit is a focus in debris flow studies since most
disasters are caused by surge impulsion and inundation.
In the previous studies, the deposition spread and run-
out distance are estimated in various ways (b
atHuRst
,
et alii, 1997; s
CHillinG
& i
veRson
, 1997; t
akaHasHi
&
y
osHida
, 1979; H
ulme
, 1974; H
aRvey
, 1984; m
izuyama
& u
eHaRa
, 1983; l
iu
& t
anG
, 1995). However, these
methods are based either on empirical relationships or
simplified dynamics largely ignoring the varieties within
surges. They make unique and certain prediction for a po-
tential event under given environment. But observations
have shown that the deposit is formed by aggradations of
successive surges that vary considerably in many ways
(m
aJoR
, 1997; v
allanCe
& s
Cott
, 1997; s
oHn
et alii,
1999). Then a debris flow involves a stochastic process
that cannot be determined by the environment conditions.
Fortunately, Jiangjia Gully (JJG) in the southwest
China exhibits a variety of debris flow appearances
and allows real-time and systemic observation (l
i
et
alii, 1983; l
iu
et alii, 2008, 2009). This paper tries to
explore the deposit features by using observation data
in the last fifty years and find the probability distribu-
tion of deposit depth.
ABSTRACT
Debris flow moves in manner of successive surges
and deposits by piling of surges. The surge occurs ran-
domly and varies in properties and magnitude. This
study explores the probability distribution of velocity
and derives the distribution of flow depth based on ob-
servations in Jiangjia Gully in the southwest of China.
The Weibull distribution is found to be well applicable
to both the velocity and depth, with parameters vary-
ing in a rather small range. Therefore, the distribu-
tion is expected to hold in general for debris flows in
different conditions and can be used to estimate the
discharge of a potential debris flow. The estimated
quantity is better than those inferred from the rainfall
at a given frequency because it incorporates both the
variation of surges and the real condition of the valley.
In conclusion, The distribution provides a more reli-
able method of risk assessment of debris flow.
K
ey
words
: debris flow; surge; weibull distribution; dischar-
ge estimation; risk assessment
INTRODUCTION
Debris flow of high density moves as surge wave
(e.g., b
laCkweldeR
, 1928; s
HaRP
& n
obles
, 1953;
P
ieRson
, 1980, 1986; t
akaHasHi
, 1991; i
veRson
, 1997;
m
aJoR
, 1997; s
auCedo
et alii, 2005; l
iu
et alii, 2008,
2009); it leaves deposit retaining configurations of the
surge, such as the lobate front and layer, lateral levee,
inverse grading, and blunt margins (n
ayloR
, 1980;
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Y. LI, P. SU & J. LIU
942
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
where P (v) is the probability of surge with velocity big-
ger than v. Statistics for some events are listed in Table 1.
As for the flow depth, it is found to be related to
velocity by a power law (l
i
et alii, 2005; 2010):
v = kh
n
This holds on average for all events of debris flow
in JJG. The coefficient k varies with channel condition
(e.g. the roughness). Besides, k varies little around 6.0
and the exponent n is about 0.40 on average (l
i
et alii,
2005). Accordingly, the flow depth, as a power function
of velocity, also satisfies the Weibull distribution (Fig.3):
P (h ) = exp (- a
H
h
bH
)
with the parameter a
H
= (k/a)
b
, and b
H
= bn. The pa-
rameters can be estimated roughly, for example, by
the average a (6.42) and b (4.11) in table 1 together
with the statistic results of k and n in Eq.(3). An esti-
mate on average is
a
H
= (k/a)
b
~ (6/6.42)
4.11
~ 0.80,
b
H
= bn ~ 4.11 x 0.4 ~ 1.60 (5)
CONFIRMATION BY OBSERVATION
The derived distribution of flow depth can be
confirmed by measurements in JJG. For operation on
Matlab, it is convenient to try the function in Eq.(2) as
P (h) = C exp (- a
H
h
bH
).
Fig. 3 displays the data points of three events,
with the fitting parameters (a
H
, bH) listed in the legend
box. The calculated results for some events are listed
in Tab. 2 and Fig. 8 presents the data in log-log plot.
FIELD OBSERVATION
Debris flow in JJG occurs at high frequency and in a
variety of appearances. Observation has continued since
1960s and a huge dataset is available now for systemic
analysis (for more information of JJG, see, e.g., l
i
et alii,
1983; d
avies
, 1990; l
i
et alii, 2003, 2004; l
iu
et alii,
2009). Each debris flow contains dozens or even hun-
dreds of surges and the deposit of a single surge looks
like a “frozen” surge and keeps the same configuration.
The photo in Fig.1 clearly shows the flowing surges and
the deposited surges on the gentle slope outside the chan-
nel. There is a remarkable similarity between surges in
motion (bright in the center) and in termination (black
and grey) (Fig. 1), which acts as the unit of deposition.
Assumed as a Bingham fluid, a surge deposits when the
shear stress is smaller than the yield strength:
τ < ρ g j h
where τ is the shear stress, ρ the density of flow, g the
gravity acceleration, j the slope gradient of the channel,
and h the flow depth (J
oHnson
, 1970; J
oHnson
& R
odine
,
1984; s
oHn
, 2000; w
anG
et alii, 2000). Due to this, each
surge retains entirety and the superposition of successive
surges make up a deposition. In a wide open slope one
can distinguish different surges by the bifurcations of the
distal ends, the lateral margins (see Fig. 2), and some-
times the overlapping wedges (e.g. s
oHn
, 2000).
DISTRIBUTION OF FLOW DEPTH
THEORETIC DERIVATION
As debris flow deposits by superposition of many
surges in a random way, we apply a probabilistic view-
point. At first we consider the distribution of the flow ve-
locity, which is the most dynamic parameter of the surge.
We find that the velocity satisfies the Weibull distribu-
tion, which in the form of exceedance probability is:
P (v) = exp (- (v/a)
b
)
(1)
Fig. 1 - Deposition of individual surges
Tab. 1 - Parameters of the weibull distribution for debris
flow velocity
(2)
(3)
(4)
(5)
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PROBABILITY DISTRIBUTION OF DEBRIS FLOW DEPTH AND ITS IMPLICATION IN RISK ASSESSMENT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
943
over surges of each event. This leads to the coinci-
dence of data points from different events (Fig. 5).
Therefore, all the events are subject to the same
It is noted that the coefficient C is exclusively near
1 (the average is about 1.04 and the standard variance
is 0.009), and the fitting curve is fine with R
2
near 1.
This confirms the validity of the Weibull distribution.
Besides, the shape parameter b
H
varies little, with av-
erage 1.85 and variance 0.21.
The scale parameter a
H
also varies little with sev-
eral exceptions. a
H
for events 890802 and 890803 are
high (5.49 and 4.21) and for events 870627 and 950715
are small (0.53 and 0.52). These abnormities corre-
spond to the fact that event with big a
H
is composed
of “shallow” surges in depth of less than 1.0m and that
event with small a
H
is composed of “deep” surges hav-
ing relatively high flow depth. As shown in Fig. 4, the
abnormal events are “off” the central data points.
More importantly, the abnormity of scale pa-
rameter can be eliminated by rescaling the depth
by h* = h/(│h
2
│/│h│), where │ │ denotes average
Fig. 3 - Probability distribution of flow depth for three events
Tab. 2 - Parameters for cumulative distribution of flow depth
Fig. 4 - Probability distribution of flow depth for debris
flows in JJG
Fig. 2 - v-h relationship (event 910715)
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Y. LI, P. SU & J. LIU
944
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Again, the value bn = 1.60 is used here to estimate
the average value. Therefore the distribution of flow
depth can be well derived from the designed velocity.
IMPLICATIONS FOR RISK ASSESSMENT
The distribution derived above provides an easy
assessment of risk. Specifically, the depth distribu-
tion can be used to evaluate the inundated area of a
potential debris flow. The inundated area is hard to
determine in practice because of the complexity of
landform and the uncertainty of the flow. Instead, it
is usually estimated by a postulated discharge or the
designed discharge for engineering structure.
In engineering design, a discharge (Q
p
) is usually
presumed by a given frequency corresponding to the
rainfall and then the discharge is used to determine the
cross-section and velocity. This methodology ignores
any variations of debris flow which might be consider-
able even under the given condition.
On the other hand, discussions above suggest that
debris flow occurs in a random way and the veloc-
ity conforms to a certain probability distribution. Be-
sides, the distribution parameters vary slightly with
events; it is possible to suppose the distribution holds
in general. This is reasonable because the velocity is
mainly determined by the fluid physics and the flow
regimes of debris flow are similar in various condi-
tions. Reports of debris flow in other areas also indi-
cate that velocity varies in the similar range, mainly
between 5 m/s and 15 m/s. This means the distribution
is generally applicable for assessment. Consider the
distribution with average parameters, i.e., a = 6.4, b =
4.1, and t = a
-b
= 0.0005 (table 1):
P (v) = exp (-t v
b
) = exp (- 0.0005 v
4.1
)
The medium velocity (i.e. with probability of
50%) is v = 5.84 (m/s), and the probability of v >9.26
is less than 1%. The 1%-possible velocity can be prop-
erly taken as the maximum of velocity in general cas-
es. Correspondingly, the maximal flow depth of 1%
possibility satisfies P (H) = exp (- a
H
H
bH
) ~ 0.01. As
a
H
~ 1.0 and b
H
~ 1.6 on average, this gives H ~ 2.6m.
Then we can derive the corresponding discharge
Q
d
= VHS
where S is the wet perimeter of the cross-section
passed by the flow and can be measured in field, and
HS gives the area of cross-section. For example, Fig.
6 shows a typical cross-section in a debris-flow chan-
nel which retains surge marks of different flow depth
distribution on average level, having almost the same
parameters. The average value of a
H
and b
H
are 1.04
and 1.75, with variance of 0.087 and 0.079, respective-
ly. This agrees well with the rough estimate of Eq.(5).
And for individual events, even better agreement can
be achieved. Consider the event 910715, for example,
k = 5.97, n = 0.42; and the Weibull parameter for veloc-
ity is a = 5.98 and b = 3.85 (Tab. 1). Then one gets a
H
= 0.99 and b
H
= 1.61. Thus the distribution parameters
are well fixed and can be easily estimated to a satisfac-
tory accuracy from the velocity probability. This is a
very admirable virtue for practice in risk assessment.
PARAMETER DETERMINATION
In order to use the distribution for risk assessment,
we should determine the parameters in general. Fol-
lowing discussions above, we assume that the shape
parameter is universally applicable and the scale pa-
rameter is related to the average value. According to
Weibull distribution, the average velocity <V> is
<V> = a Γ(1 + 1/b)
where
Γ (1 + x) = xΓ (x) (x > 0) is the Gamma func-
tion. For the case of JJG, 1/b is less than 1/3 (Tab. 1),
then <V>= a/bΓ(1/b) >0.90a. Thus the scale parameter
a can be estimated by the average velocity at the accu-
racy of 90%. For a valley to be assessed, we suppose
the expected velocity is β times the maximal veloc-
ity of JJG, then the same factor β also applies to the
average value of the expected velocity. Thus the scale
parameter for velocity distribution is βa.
The parameters for flow depth distribution can be
similarly derived. Corresponding to the velocity, the
average depth is
<H> = βa/kΓ(1 + 1/bn) = βa/(kbn)Γ(1/bn) ~ βa/k
Fig. 5 - Cumulative distribution of normalized flow depth
(6)
(7)
(8)
(9)
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PROBABILITY DISTRIBUTION OF DEBRIS FLOW DEPTH AND ITS IMPLICATION IN RISK ASSESSMENT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
945
1) Debris flow moves in manner of separate surges
and the surge acts as the unit of flow motion and
deposition. This is determined by the nature of the
fluid but not by the environmental factors;
2) Debris flows in JJG originate from various sources
and under different rainfalls; therefore they repre-
sent a variety of physical conditions. Moreover,
the surges cover a wide spectrum of motion regi-
mes. In other words, debris flow in JJG presents
properties and performances of debris flows in
various regions and conditions;
3) The distribution parameters fall into a small ran-
ge; this means that individual events conform to
the same rule despite their varieties of origins.
Therefore the probability distribution is expected
to be generally applicable for debris flow in various re-
gions and conditions, only with small variation of scale
parameters that doesn’t change the form of distribution.
Discharge estimated through the distribution is
expected to be more reliable because it incorporates
the living performance of debris flow surge other than
derives from the indirect conditions of debris flow,
such as the background and the rainfall.
Additionally, although there are rare valleys like
JJG that develop debris flows with such a high fre-
quency and variety, debris flow is probabilistic even if
only one event falls in a valley. Thus the probabilistic
scenario we get from JJG might as well provide a pro-
totype for assessing debris flows in different valleys.
ACKNOWLEDGEMENT
The authors are grateful to the financial supports
from the Knowledge Innovation Program of the Chi-
nese Academy of Sciences (Grant No. KZCX2-YW-
Q03-5-2) and the National Science Foundation of
China, Grant No. 40771010.
and hence different inundation areas. Eq.(7) thus de-
termines the discharge by probability estimation and
field condition. In other words, we can derive the dis-
charge at a certain probability from field observation.
This differs from the postulated discharge in that: 1)
the postulated discharge is estimated by combining
the rainfall of a given frequency and the material con-
centration (l
iu
et alii, 2009). But in reality, the same
rainfall doesn’t necessarily cause the same discharge,
and the discharge doesn’t necessarily concur with the
rainfall; and 2) the derived discharge incorporates the
variation of dynamic parameter and the real condition,
and thus may be more reliable and practicable.
CONCLUSIONS AND DISCUSSIONS
Deposition of debris flow actually results from
piling of separated surges. As the surge varies remark-
ably and randomly, we employ the probability distri-
bution to set an overall view of the process.
Using observation data in the valley of JJG we’ve
built the Weibull distribution for velocity and then de-
rived the similar distribution for flow depth. Then the
distribution provides a method to estimate the discharge
of the potential debris flow at a certain probability.
Although the velocity distribution is derived from
JJG, its universality is justified by the following reasons:
Fig. 6 - A typical cross-section with debris flow surge
marks of different flow depth
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