# IJEGE-11_BS-Li-Su-&-Liu

*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-102*

**PROBABILITY DISTRIBUTION OF DEBRIS FLOW DEPTH**

**AND ITS IMPLICATION IN RISK ASSESSMENT**

deriving parameters of debris flow as fluid of visco-

plasticity (m

*et alii*, 1996).

In the previous studies, the deposition spread and run-

out distance are estimated in various ways (b

*et alii*, 1997; s

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

*et alii*,

that cannot be determined by the environment conditions.

and allows real-time and systemic observation (l

*et*

*alii*, 1983; l

*et alii*, 2008, 2009). This paper tries to

in the last fifty years and find the probability distribu-

tion of deposit depth.

**ABSTRACT**

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**

*et alii*, 2005; l

*et alii*, 2008,

surge, such as the lobate front and layer, lateral levee,

inverse grading, and blunt margins (n

*Y. LI, P. SU & J. LIU*

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

ger than v. Statistics for some events are listed in Table 1.

*et alii*, 2005; 2010):

*v*=

*kh*

*n*

*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

*et alii*,

of velocity, also satisfies the Weibull distribution (Fig.3):

*P*(

*h*) = exp (-

*a*

*h*

*b*H

*a*

*k*/

*a*)

*bn*. The pa-

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*

*k*/

*a*)

*bn*~ 4.11 x 0.4 ~ 1.60 (5)

*CONFIRMATION BY OBSERVATION*

Matlab, it is convenient to try the function in Eq.(2) as

*P*(

*h*) =

*C*exp (-

*a*

*h*

*b*H

*a*

*b*H) listed in the legend

in Tab. 2 and Fig. 8 presents the data in log-log plot.

**FIELD OBSERVATION**

1960s and a huge dataset is available now for systemic

analysis (for more information of JJG, see, e.g., l

*et alii*,

*et alii*, 2003, 2004; l

*et alii*,

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*

*ρ*the density of flow,

*g*the

gravity acceleration,

*j*the slope gradient of the channel,

and

*h*the flow depth (J

*et alii*, 2000). Due to this, each

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

**DISTRIBUTION OF FLOW DEPTH**

*THEORETIC DERIVATION*

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*

*Fig. 1 - Deposition of individual surges*

*Tab. 1 - Parameters of the weibull distribution for debris*

*flow velocity*

**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*

dence of data points from different events (Fig. 5).

*C*is exclusively near

is 0.009), and the fitting curve is fine with

*R*

Besides, the shape parameter

*b*

*a*

*a*

are small (0.53 and 0.52). These abnormities corre-

spond to the fact that event with big

*a*

event with small

*a*

abnormal events are “off” the central data points.

by

*h** =

*h*/(│

*h*

*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)*

*Y. LI, P. SU & J. LIU*

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

*bn*= 1.60 is used here to estimate

depth can be well derived from the designed velocity.

**IMPLICATIONS FOR RISK ASSESSMENT**

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.

*Q*

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.

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*

*P*(

*v*) = exp (-

*t v*

*b*

*v*

*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*

*b*H

*a*

*b*

*H*~ 2.6m.

*Q*

*VHS*

*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

parameters. The average value of

*a*

*b*

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*

*b*

tory accuracy from the velocity probability. This is a

very admirable virtue for practice in risk assessment.

**PARAMETER DETERMINATION**

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)*

*Γ (1 + x) = xΓ (x) (x > 0)*is the Gamma func-

*1/b*is less than 1/3 (Tab. 1),

then <

*V*>=

*a*/

*b*Γ(1/

*b*) >0.90

*a*. 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*.

average depth is

*H*> =

*βa*/

*k*Γ(1 + 1/

*bn*) =

*βa*/(

*kbn*)Γ(1/

*bn*) ~

*βa*/

*k*

*Fig. 5 - Cumulative distribution of normalized flow depth*

**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*

deposition. This is determined by the nature of the

fluid but not by the environmental factors;

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;

the same rule despite their varieties of origins.

Therefore the probability distribution is expected

gions and conditions, only with small variation of scale

parameters that doesn’t change the form of distribution.

the living performance of debris flow surge other than

derives from the indirect conditions of debris flow,

such as the background and the rainfall.

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**

nese Academy of Sciences (Grant No. KZCX2-YW-

Q03-5-2) and the National Science Foundation of

China, Grant No. 40771010.

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

*et alii*, 2009). But in reality, the same

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**

ably and randomly, we employ the probability distri-

bution to set an overall view of the process.

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.

*Fig. 6 - A typical cross-section with debris flow surge*

*marks of different flow depth*

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