Assessment of Risk due to Debris Flow
Events: A Case Study in Central Taiwan
GWO-FONG LIN1,w, LU-HSIEN CHEN2and JUN-NAN LAI1
1
Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan; 2
Department of Information Management, Diwan College of Management, Madou, Tainan 72141, Taiwan
(Received: 30 December 2003; accepted: 9 August 2005)
Abstract. In this paper, based on the concepts of uncertainty and reliability analyses, a method used for assessment of risk due to debris flow events is proposed. First, procedures for obtaining the configuration of debris-flow fans are presented. Then, the parameters affecting the configuration of debris-flow fans are identified and their corresponding means and stan-dard deviations are derived. Finally, the proposed method is applied to the Shih-Pa-Chung Creek in central Taiwan. The expected deposition thickness at any point in the deposition area is computed and then the contours of risk for the 50-year and 100-year events are constructed. Regarding the expected deposition thickness, it is found that the closer the distance from the canyon mouth, the larger the debris-flow thickness becomes. The results also show that the contours of risk are of the shape of an ellipse similar to the shape of deposition area, and the risk at a point decreases with increasing distance of that point from the canyon mouth. In addition, when the return period of rainfall event is fixed, the variation in risk decreases as the distance from the canyon mouth increases. For the assessment of risk due to debris flow events, the proposed method is recommended as an alternative to the existing methods, be-cause the influence of all the uncertainty of the parameters is considered.
Key words: risk assessment, debris flow, deposition area, reliability, uncertainty
1. Introduction
Debris flows, which are sometimes referred as mudslides, mudflows, lah-ars, or debris avalanches, are common types of fast-moving landslides. These flows generally occur during periods of intense rainfall or rapid snowmelt. They usually start on steep hillsides as shallow landslides. Initiation of debris flows requires loose rock and soil deposits, steep slop, and free flowing water from snowmelt or a rainstorm. Debris flows from many different sources can combine in channels where their destructive power may be greatly increased. They continue flowing down hills and through channels, growing in volume with the addition of water, sand,
w
boulders, trees, and other materials. When the flows reach canyon mouths or flatter ground, the debris spreads over a broad area and the debris-flow fan forms.
Taiwan is a mountainous island where two-thirds of area located in mountainous zones. Unfortunately, its geological condition is weak, and hence during heavy rainfalls, the colluvium is easily weakened, which often leads to a debris flow. Many severe debris flows have happened in Taiwan (Hung, 1996; Cheng et al., 1997, 2000). For example, the 1990 Tong-Mang Valley event (located in eastern Taiwan) that claimed 35 lives, and the 1996 Chen-Yu-Lan River event (located in central Taiwan) that claimed 18 lives. The debris flows often cause serious loss. Therefore, how to reduce the debris-flow hazards becomes the most important task in Taiwan. The processes of debris flows are complex and reliable modification of it re-quires a degree of understanding. An easy method to reduce debris flow disasters is to remove people and their assets from areas likely to be af-fected by debris flows (Davies, 1997). Unfortunately, this is impossible ow-ing to the limited space, as well as social and legal difficulties. On the other hand, a real-time prediction and warning system is helpful (Sorensen, 2000). However, it is inevitable that people face debris flow disasters. The assessment of risk due to debris flow events prior to actual disasters, there-fore, becomes an effective method to reduce the debris-flow hazards.
Before assessing debris-flow hazard, one should first know the configu-ration of debris-flow fan. That is, one should know how long, how wide and how thick the debris flow can deposit, as well as how much volume and in what zones the debris flow deposits. The process of formation and the shape of debris-flow fan have drawn attention of many researchers such as Takahashi (1991), Hungr (1995), Iverson (1997), Shih et al. (1997), Fannin and Wise (2001), and Wilkerson and Schmid (2003). Takahashi (1991) proposed empirical formulas for the probable maximum length and thickness of the debris-flow deposits. Major and Pierson (1992) made experimental analysis of fine-grained slurries. Shieh and Tsai (1997) con-structed the relationships among the maximum length, width and thickness based on the experimental data. Shih et al. (1997) studied the grading of risk for hazardous debris-flow zones. Lin et al. (2002) applied geographic information system (GIS) techniques for assessing debris-flow hazards. Clague et al. (2003) used the peak discharge as input into an empirical equation to estimate the volume of debris flows. D’Ambrosio et al. (2003) presented a cellular automata model for simulating debris-flow phenom-ena. Liu and Lei (2003) presented a new method that can estimate debris flow hazard from gully density, mean annual rainfall and percentage of cultivated land on steep slope.
So far, the delineation of the debris-flow deposition area is limited to a deterministic approach. That is, existing methods ignore the influence of
the parameters on the probable maximum length, width and thickness. However, the influence factors of debris-flow deposition are numerous and the empirical, self-empirical or theoretical models that have been developed are too simple. To build a complete and reasonable model for delineating debris-flow hazards is not easy. Hence, uncertainty and reliability analyses (Yen and Tung, 1993) should be used. Reliability analysis is increasingly used in applications of hydraulic and hydrologic engineering, such as groundwater (Hamed et al., 1996), debris flows (Archetti and Lamberti, 2003; Lin et al., 2004), water resource management (Yen and Tung, 1993), watershed water quality model (Bobba et al., 1996), and selective index for regional flood frequency analysis (Lin and Chen, 2003).
There are two objectives in this paper. One is to find the expected 3-D distribution of the debris flow and the other is to compute the risk based on the concepts of uncertainty and reliability analyses. We define ‘risk’ as the probability that the thickness of the deposition area is larger than a certain height herein. First, the configuration of debris-flow fan is intro-duced. Furthermore, the mean and variance of thickness at any point are derived. Finally, the actual application is performed. The expected deposi-tion thickness at any point in the deposideposi-tion area is computed and the con-tours of risk are constructed.
2. Configuration of Debris-Flow Fan
To compute the configuration of debris-flow fan, the maximum deposition length, width and thickness need to be known. Procedures for obtaining the configuration of debris-flow fan are presented as below. Following the experiments of Shieh and Tsai (1997), let x is the distance from the canyon mouth, and then the thickness Zc (see Figure 1) in the primary debris-flow direction, which varies with x, can be described by
Zc ¼ Zmaxexp 1 2C2 L x Lmax 2 " # ð1Þ
where Zmax is the maximum deposition thickness that is at x=0 (Figure 1),
Lmax is the maximum deposition length, and CL is a coefficient varying from 0.39 to 0.40. As to the estimation of Lmax, the formula proposed by
Takahashi (1991) has been widely used (see Appendix A). At any point (x, y), the deposition thickness can be written as
Zxy ¼ Zcexp 1 2C2 B y b 2 ð2Þ
where b is the deposition width at x (see Figure 2) and CB is a coefficient varying from 0.20 to 0.21. The Zmax in Equation (1) and b in Equation (2) can be obtained as follows. According to Figure 1, the maximum deposi-tion thickness Zmax can be written as
Zmax¼ Lmaxtanðh hdÞ ð3Þ
where h is the longitudinal slope of debris-flow deposits (degree) and hd is the bed slope of debris-flow fan downstream of the end of the flow channel (degree).
According to the experimental results of Shieh and Tsai (1997), the shape of the deposition area (Figure 2) can be described by
b Bmax 2 þ x Lmax 2 ¼ 1; 0 x Lmax ð4Þ
where Bmax is the maximum deposition width, and x, b and Lmax are as defined earlier. According to Equation (4), the width at any distance x from the canyon mouth is
b¼ Bmax 1 x Lmax 2 " #0:5 ð5Þ
3. Assessment of Risk due to Debris Flow Events
The methodology used for assessment of risk due to debris flow events is briefly presented in this section. When performing an assessment of risk due to debris flow events, one expects to know the risk, i.e. the probability that the thickness at any point of the deposition area is more than a certain height. The probability that the thickness Zxy exceeds a certain height Z* is referred to as risk r¢:
r0¼ Prob½Z Zxy ¼ Prob½SM 0 ð6Þ
where safety margin SM=Z*) Zxy. Equation (6) can be further written as r0¼ Prob½U ðlSM=rSMÞ ¼ 1 FUðlSM=rSMÞ ð7Þ
where U¼ ðSM lSMÞ=rSM is the standardized normal random variable,
FU (u) is the cumulative distribution function of the standardized random variable U, and lSM and rSM are the mean and the standard deviation of
SM, respectively. The lSM can be obtained from lSM¼ E Z Zxy
¼ lZ lZxy ð8Þ
where E is the expectation operator, and lZ and lZxy are the means of Z*
and Zxy, respectively. When Z* and Zxy are independent, r2SM is given by
r2SM¼ r2Zþ r2Zxy ð9Þ
where r2
Z and r2Zxy are respectively the variance of Z* and Zxy. In this paper, Z* is set equal to a certain height, i.e. Z* is deterministic. Thus we need to know lZxy and r
2
Zxy to calculate lSM and r
2
SM according to
Equa-tions (8) and (9). Once lZxy and r2
Zxy are obtained, r¢ can be computed
from Equation (7). In Appendix B, the mean and variance of Zxy are derived.
4. Application and Discussions
The Shih-Pa-Chung Creek susceptible to debris-flows in central Taiwan is chosen as the study area in this paper. Figure 3 shows the study area and the location of the Wan-Hsiang rain gauge. This creek is a tributary of the Chen-Yu-Lan River that flows into the Chou-Shui River, the longest river in Taiwan. The Shih-Pa-Chung Creek is incised along the Ti-Li Fault and close to the Chen-Yu-Lan Fault, so that the geology and topography cre-ate a environment prone to debris flows. In the recent years, three major debris flows occurred in this study area in 1986, 1987, 1996 (Cheng et al., 2000). Therefore, the Council of Agriculture (1996) classified the Shih-Pa-Chung Creek as a hazardous debris-flow stream.
In this application, at first we need to estimate the means and vari-ances of the upstream and downstream bed slopes of the debris-flow path, drainage area, the length of overland flow path, the length of chan-nel, and the elevation difference between the inlet and outlet of the main channel. In this application, based on the topographic map with a scale of 1:10,000, well-trained persons are asked to measure these variables. For each random variable, a set of readings from different persons is then formed from which the mean and variance can be estimated. Table I presents the means and standard deviations of the aforemen-tioned geometric variables, and Table II presents the means and standard
deviations of the hydrologic variables and the debris-flow variables for two different return periods (50 and 100 years). The variables in Table II are estimated using the formulas in the literature. One can refer to Lin et al. (2004) for the details of the formulas. The key factor in the estima-tion of the variables is rainfall intensity. As to rainfall, in total we use 43 annual rainfall observations (1960–2002) to find the 50- and 100-year design rainfall intensities. In addition, Table III summarizes the means
Table I. Means and standard deviations of geometric variables.
Variable Mean Standard deviation
Bed slope of canyon hu(degree) 14.0 3.5
Drainage area A (ha) 210.0 10.5
Table II. Means and standard deviations of the hydrologic variables and the debris-flow variables for two different return periods.
Variable Return period T (years)
50 100
Mean Std. dev. Mean Std. dev.
Peak water discharge Qw(m3/s) 61.0 14.6 66.5 14.9
Concentration of debris-flow deposits C* 0.65 0.03 0.65 0.03
Longitudinal slope of debris-flow deposits h (degree)
8.0 0.8 8.0 0.8
Equilibrium debris-flow concentration Cdu 0.3 0.2 0.3 0.2
Debris-flow discharge QD(m 3
) 125.9 35.4 137.3 37.7
Mean upstream debris-flow velocity Uu(m/s) 9.2 6.5 9.7 6.8
Debris-flow volume V (m3) 39,548 29,198 43,127 31,644
Table III. Means and standard deviations of the maximum deposition length, width and thickness for two different return periods.
Variable Return period T (years)
50 100
Mean Std. dev. Mean Std. dev.
Maximum deposition length Lmax(m) 97.9 53.9 107.1 58.9
Maximum deposition thickness Zmax(m) 3.4 1.2 3.7 1.3
and standard deviations of the maximum deposition length, width and thickness on the debris-flow fan under two conditions of return period (50 and 100 years). For the case studied herein, Table III shows that the expected maximum deposition length and thickness increase with increas-ing return period of rainfall intensity, whereas the expected maximum deposition width decreases with increasing return period.
The expected 3-D pictures of the debris-flow fan in different return periods are computed. Figure 4 gives the expected deposition in three-dimensional form for the 50-year event. In a like manner, Figure 5 shows the result for the 100-year event. The closer the distance from the canyon mouth is, the larger the debris-flow thickness becomes.
The other objective of this paper is to compute the probability that the thickness at any point of the debris-flow fan is more than a certain height. In this paper, the certain height is taken as 3 m (the average height of the first floor in Taiwan). That is, one proceeds to compute the probability that the thickness at any point is more than 3 m using the first-order sec-ond moment method. Figures 6 and 7 represent the contours of risk, i.e. probability that thickness is more than 3 m, for the 50- and 100-year return period rainfall events, respectively. As shown in Figures 6 and 7,
Figure 4. The expected three-dimensional form of the deposition for the 50-year event.
one can find that the contours of probability are of the shape of an ellipse similar to the shape of deposition area. The probability that the thickness at a point is more than 3 m decreases as the distance of that point from the canyon mouth increases. In addition, the variation in risk decreases with increasing distance from the canyon mouth when the return period of rainfall event is fixed.
5. Summary and Conclusions
The primary purpose of this paper is to find the expected 3-D distribution of the debris flow and to compute the probability that the thickness at any point of the deposition area is more than a certain height. First, the configuration of debris-flow fan is described. Then, the parameters affect-ing the probable maximum length, width and thickness are identified and their corresponding means and standard deviations are derived. In addi-tion, actual application of the proposed methodology is performed. As to the computation of the probability that the thickness at any point of the deposition area is more than a certain height, the first-order second moment method is used. It is found that the risk at a point decreases as the distance of that point from the canyon mouth increases. Regarding the expected deposition thickness, the closer the distance from the canyon
Figure 5. The expected three-dimensional form of the deposition for the 100-year event.
mouth is, the larger the debris-flow thickness becomes. In fact, the actual topography is so complex that it will be difficult to understand the charac-teristics and conditions for debris flows. For the purpose of assessing risk due to debris flow events, a preliminary study was carried out in this paper (i.e. assume the geometry of the debris flow fan to be ellipsoid form). The proposed method is expected to be useful for assessment of risk due to debris flow events.
Appendix A: The Maximum Deposition Length and the Maximum Deposition Width
The probable maximum deposition length Lmax from the end of the flow channel can be estimated from (Takahashi, 1991)
Lmax ¼ Uu cosðhu hdÞ 1 þ½ðsqÞC2ðsqÞDEKaþqcos hu DEþq ½ ghu U2 u 2 ðsqÞgCDEcos hdtan a ðsqÞCDEþq g sin hd ðA1Þ
where hu is the bed slope of the flow channel (degree), hd is the bed slope of debris-flow fan downstream of the end of the flow channel (degree), g is the acceleration of gravity (m/s2), s is the density of gravel (g/cm3), q is the density of water (g/cm3), hu is the average debris-flow depth in the flow channel (m), Uu is the cross-sectional mean velocity of debris flow in the flow channel (m/s), Ka is the coefficient of active earth pressure, a is the angle of dynamic friction (degree), and CDE is the equilibrium debris-flow concentration. The values of parameters hu, hd, s, hu, Uu, a and CDE origi-nate from measurement methods, field data, or the expert knowledge of the investigators.
Figure 7. Contours of probability that thickness is more than 3 m for the 100-year event.
The probable maximum deposition width at the canyon mouth (end of the flow channel) can be estimated from (Shieh and Tsai, 1997)
Bmax ¼
V j Zmax Lmax
ðA2Þ
where V is volume of debris flow (m3), Zmax is the maximum deposition thickness (m), and j is a coefficient varying from 0.220 to 0.235.
Appendix B: Mean and Variance of Thickness at Any Point in the Deposition Area
The mean value first-order second moment (MFOSM) method (Yen and Tung, 1993) is adopted to find the mean and variance of thickness at any point in the deposition area. The deposition thickness Zxyat any point (x, y) is given by Equation (2). Zc, CB, and b in Equation (2) are regarded as random variables. Hence, the mean ( Zxy) and variance (sZxy
2 ) of Z xyare Zxy ¼ Zcexp 1 2 C2B y b 2 " # ðA3Þ s2Zxy ¼ exp 1 2 C2 B y b 2 " # ( )2 s2Z cþ y2 b2C3 B Zcexp 1 2 C2 B y b 2 " # ( )2 s2C B þ y 2 b3C2 B Zcexp 1 2 C2 B y b 2 " # ( )2 s2b ðA4Þ
As previously indicated, CB ranges from 0.20 to 0.21. It is reasonable to assume that CB is equally likely to take any value over that range. Hence, one can refer to a general textbook of statistics to find the mean and vari-ance of the uniformly distributed random variable CB. Moreover, Zc and
sZc2 are the mean and variance of the debris-flow thickness in the primary flow direction, respectively. According to Equation (1), one can obtain Zc
and sZc2 as Zc ¼ Zmaxexp 1 2C2L x Lmax 2 " # ðA5Þ
s2ZC ¼ fexp½ 1 2 C2 L ðx Lmax Þ2g2s2Z maxþ f Zmaxx2 C2 LL3max exp½ 1 2 C2Lð x Lmax Þ2g2s2L max ðA6Þ In Equations (A5) and (A6), the means and variances of Zmax and Lmax must be computed. One can refer to Lin et al. (2004) for the mean ( Lmax)
and variance ( s2L
max) of Lmax. According to Equation (3) in which Lmax and
h are random variables, the mean ( Zmax) and variance ( s2Zmax) of Zmax can
be written as
Zmax¼ Lmaxtanðh hdÞ ðA7Þ
s2Z
max ¼ ½tanðh hdÞ
2s2
Lmax þ ½Lmaxsec
2ðh h
dÞ2s2h ðA8Þ
As shown in Equations (A7) and (A8), the mean and variance of h are needed. In general, h is between 6 and 10 degrees according to field investi-gation in Taiwan (Shieh and Tsai, 1997) and it is considered herein to be uniformly distributed over that range.
In Equations (A3) and (A4), terms left to be found so far are the mean ( b) and variance (sb2) of b. According to Equation (5), b and sb2 can be obtained as b¼ Bmax 1 x Lmax 2 " #0:5 ðA9Þ s2b¼ 1 x Lmax 2 " #0:5 8 < : 9 = ; 2 s2B maxþ Bmax 1 x Lmax 2 " #0:5 x2L2max 8 < : 9 = ; 2 s2L max ðA10Þ As to the mean ( Bmax) and variance ( s2Bmax) of Bmax in Equations (A9) and
(A10), one can refer to Lin et al. (2004).
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