Flood Risk andClimate Change
Hokkaido
WP2 Bank failure probability

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Flood Risk andClimate Change
Hokkaido
WP2 Bank failure probability
Final report

Authors
Fum ihiko, UEMURA (Docon Co.,Ltd)
Rongen, Guus (HKV consultants)

PR3983January 2022

Table of contents
1

Introduction

1

1.1
1.2

Objective
Structure report1
1

2

Method3

2.1

Mechanism of dike failure

3

2.2

Overview of the our approach4

3

Target basin

5

4Inner slope failure mechanism

6

4.1

Dutch methods for grass revetment failure

6

4.2

Cumulative overtopping dam age for the inner slope

6

4.3

Fragility curves: failure probability given water level

85Sources of uncertainty

9

5.1

Water levels

95.2

Bank Height

11

5.3

Critical flow velocities

12

5.4

Com bining the uncertainties

13

6

Bank failure probability

15

6.1

Calculate failure probabilities

15

6.2

Com bine segments to sections

15

6.3

Segment probabilities

17

6.4

Com bining section probabilities per river branch

17

6.5

Com paring the calculated failure probabilities

18

7

Flood probabilities

19

7.1

Discharge statistics

19

7.2

Flood probability

20

8

Sensitivity analysis

22

8.1

Uncertainty in bank height, crest level and rating curve

22

8.2

Hydrograph shape

23

8.3

Concluding

25

9

Discussion

26

9.1

Failure Probability

26

9.2

Flood probability

26

10Conclusions and Recommendations

29

11

References30

1

Introduction
This report gives an overview of the activities from work package 2 (WP2).
This work package is one of the three work packages that gives insight in theflood risk for Hokkaido and specifically the case study Obihiro. WP1 gives
insight in the im pact of clim ate change on extreme water levels, WP2
focusses on probability of levee failure.
The probability of flood occurrence consists of the effects of the probability ofwater level exceedance and the probability of dike failure. The m ethod of
calculating and analyzing the probability of water level exceedance is
described in the WP1 report. In this report, only the results of the calculat ionof the probability of water level exceedance are described.
For the probability of dike failure, a fragility curve is calculated by focusing on
the m echanism of levee failure due to water overflow. Uncertainty is includedin the fragility curve. Uncertainty takes into account the relationship between
water level and flow rate, the height of the planned and actual levees, andthe condition of the grass on the surface of the levees.

1.1

Objective
The objective of work package 2 is to determ ine the flood prob ability for the
Obihiro case and give insight in the uncertainties in bank failure probability
calculation.

1.2

Structure report
The report starts with an overview of the targeted failure m echanisms andthe approach taken to analyze them probabilistically (Chapter 2). Chapter 3
describes the target basins and districts, and explains the concept of
longitudinal classification of assessments. Chapter 4 describes the physicalconditions that cause failure and how to create a fragility curve. Chapter 5
describes the uncertainties included in the fragility curve. These are: waterlevel (5.1), bank height (5.2), and inner slope condition (5.3). The failure
probability is calculated in Chapter 6. Here, a m ethod is also proposed to take
into account the upstream probability (6.2) In Chapter 7, the flood probability
is calculated, including the flow statistics provided by WP1. In Chapter 8,
sensitivity analysis was conducted on the uncertainty included in the fragility
curve and the shape of the hydrograph. Finally, the c haracteristics of theresults obtained from this method and future prospects are discussed(Chapter 9), and conclusions are provided in Chapter 10.

PR3983 • January 2022

1

2

MethodIn flood control planning in Japan, calculations are based on the condition
that a breach will occur if the water level exceeds the planned water level.
The probability is evaluated as 0 or 1, whether the water level will be
exceeded or not. However, the actual phenomenon of breaching is that thewater level m ay reach the planned water level but not breach, or the waterlevel m ay not reach the planned water le vel but breach occurs. This is due tovarious factors such as the m echanism of levee failure, strength of the levee,
and tem poral changes in hydraulic conditions. Probabilistic assessment oflevee failure is essential for the correct assessment of flood risk.
In the Netherlands, flood risk has been assessed based on the probability oflevee failure using a fragility curve, which has been introduced into policy.
In this study, the Dutch m ethod of calculating the probability of breaching isapplied to the Tokachi River basin, and a m ethod of creating a fragility curve
is proposed, with som e m odifications based on the characteristics of floods inJapan.

2.1

Mechanism of dike failure
There are several factors that can cause levee breaches. In Japan, the factors
are generally divided into overflow, erosion/ scour, and seepage. In 2019,
typhoon No. 19 caused floods in Japan and breaches of levees occurred inm any places, including 14 on rivers m anaged by the national government
and 128 on rivers m anaged by prefectural governments. According to theresults of a survey on the causes of breaches, overflowing water was them ain cause of 86% of the breaches [6]. In this study, the probability of levee
breakage was calculated for overflowing water, which is one of the m ost
frequently reported causes of levee breakage in Japan.

Main factors for levee failure •by Typhoon No. 19, 2019, Japan
piping1%
erosion9%

unknown4%

n = 142In the case of the floods caused
by Typhoon No. 19 in Japan,
levee failures occurred in 14
national rivers and 128•

prefectural rivers.
The m ain factor for the failure isovertopping of water, which

Figure 1
Causes of the levee
failure caused by
Typhoon No. 19
2019, in Japan .

PR3983 • January 2022

overtopping
86%

accounts for 86%.
Based on Reference [6].

3

2.2

Overview of the our approach
The m ethod applied in this study is based on VNK2 [8] and the Dutch m ethodfor dike safety assessment, called BOI [9]. Adjustm ents have been m ade to
som e parts of the m ethod:•

Only the failure m echanism overtopping was considered, as it isconsidered the m ost im portant for Obihiro. We applied a cum ulative
dam age approach, to take into account the duration of overtopping.

•Not only the probabilities of the peak discharges are considered, butalso the shape of the hydrograph. A high water level with a long
duration gives a higher failure probability, as it can cause m ore
dam age to the revetment.

•

When calculating the total failure probability for the area, we use atailored approach to take dependencies between different dikesections into account.
1.

Failure probabilities are combined conditional to thedischarge, as different sections would fail during the sam ehigh discharge conditions.

2.

For com bining segments into section probabilities, them axim um segment failure probability per discharge is used,
hence, the segments are considered dependent.

3.

When com bining sections to river probabilities, the reducing
effect of potential failure of upstream sections is taken into
account.

The definition of segm ents and sections is explained in chapter 3.

WP2

Hydrograph

Seleted by WP3Water level

H-Q relation

Overflow

Levee height

Velocity

at slope

Damge index
failureFigure 2
Flow of calculation ofbreach probability
considering
uncertainty.

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Uncertainty

Generation of random variables
following a normal distribution

Condition of grass

Repeat N times
Number of failure judgments (n) / N

levee failure probability

4

3

Target basin
In this study, the levees of the Tokachi River and the Satsunai River adjacentto the Obihiro urban area are evaluated. An overview of this area is shown
below.
Data are collected at intervals of about 0.2 km in the target rivers , indicatedwith the circles in the figure. In this study, the levee segments are evaluatedat the sam e interval. The sections (i.e., a collection of segments) are shownwith different colors. The segment within the section that is considered in
WP3, is shown with a black cross. The KP num ber for these breach locations
are shown as well.

Figure 3
Overview of the KPlocations for which
Qh, bank height anduncertainty data are
available.

For clarity: Transects are drawn with an interval of 0.2 km along the river.
These transects are denoted with KP X, with X being the longitudal distance.
Data, such as rating curves and flood scenarios, are collected for eachtransect. The dikes are evaluated at each transect. The dike segment
represents the part of dike related to the transect. As the consequences of abreach are likely sim ilar for nearby transects, dike sections are defined,m ostly consisting of 5 to 10 dike segm ents. The dike sections are denotedwith the location of the segment that is used to calculate the flood
consequences. For exam ple, the brown section in the figure is denoted with
KP59.6, which is the segment representative for the consequences of abreach at any point along the brown line.

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5

4

Inner slope failure
mechanism
In this study, we focused on the m echanism of levee failure caused by
overflowing water, and adopted a m ethod to determine the occurrence oflevee failure based on the dam age caused by erosion of the levee slope by
overflowing floodwaters.

4.1

Dutch methods for grass revetment failure
In the Netherlands a cum ulative damage approach is used to calculate
revetm ent failure on the outer slope of the dike . On the inner slope a critical
overtopping volume (without a duration aspect) is used . However, the criticalvalues are derived from a cum ulative damage approach, sim ilar to the outerslope. In this study, we apply the cum ulative overtopping approach to theinner slope as well. This m eans that during a storm event each overtopping
wave adds a bit of dam age to the total. When a certain threshold value is
exceeded, the dike fails. We can write this in a form ula as follows:𝑁

𝐷 = ∑ max [ (𝛼𝑀(𝛼𝑎 𝑈𝑖 )2 − 𝛼𝑆𝑈𝑐2); 0] ⋯ 1
𝑖=1

In which 𝐷 is the cum ulative dam age, 𝑁 is the num ber of waves. 𝛼𝑀 and 𝛼𝑆
are the m odel factor to take objects on the inner revetment into account (𝑀for load, 𝑆 for strength), for exam ple trees that locally lead to m ore erosion.
𝛼𝐴 is the factor for speeding up acceleration of the flow on the downward
slope. 𝑈𝑖 is the m axim um depth average flow velocity for wave 𝑖. 𝑈𝑐 is thecritical flow velocity. Flow velocities larger than this critical level will cause
dam age to the revetment.
A problem with the approach from Equation 1, is that it is aim ed at waves
and not a continuous flow. There is no tim e com ponent in the equation, justthe num ber of waves and the m axim um flow velocity in each of the waves.
The num ber of waves can be translated to a tim e period, but a continuous
flow cannot.

4.2

Cumulative overtopping damage for the inner slope
Another cumulative damage approach is described in [7]. They m ake atranslation is between the overtopping/discharge towards flow velocities andduration. The general steps are as follows:

PR3983 • January 20226

Based on the water level above the crest of the levee, the overtopping
discharge can be calculated from the flow velocity on the crest of the levee:𝑢0 = √𝑔ℎ0 ⋯ 2
𝑞 = 𝑢0ℎ0 ⋯ 3This can be refined by using overflow form ulas for (broad crested) weirs, but
for sim plicity we will stick with these form ulas for now.
As the water flows over the dike, it will accelerate on the downward slope.
Based on the shallow water equation simplified to steady flow, the term inal
flow velocity can be calculated based on the slope and friction coefficient of
the levee slope, as presented by Dean et al. (2010):
18𝑔𝑞sin𝛼 3
𝑢∞ = () ⋯4𝑓This flow velocity can be compared to a critical flow velocity:33
𝐸𝑊 = 𝐾𝑊(𝑊𝑖 − 𝑊𝑐 )𝑡𝑖 = 𝐾𝑊 𝛽𝑊 (𝑢𝑚,𝑖− 𝑢𝑐,𝑊)𝑡𝑖 ⋯ 5

The 𝑊 represents the physical quantity ‘work’, hence the power 3 for which
the velocities above the critical flow velocities contribute to the damage.
𝐸𝑊 /𝐾𝑊𝛽𝑊 is calculated to be 0.492 ⋅ 10 6 for a good quality grass cover. If the3
3
flow velocities are of such a m agnitude that (𝑢𝑚,𝑖− 𝑢𝑐,𝑊)𝑡𝑖 exceeds this value,
the revetm ent fails. The critical flow velocity is calibrated on 1.80 m /s, which
is low com pared to the wave-oriented approach m entioned before.
With this approach, we can calculate the dam age for all tim e steps. All
different hydrograph shapes can be used in this way. The disadvantage ofthis m ethod is that it does not take into account the m ore recent acquired
knowledge on revetment strength. The m ethod does however differentiate
between grass cover qualities and provides standard deviations for the criticalflow velocities.
An exam ple for a hypothetical hydrograph shape and a good grass cover isshown in the figure below. The left figure shows the evolution of the waterlevel. As soon as the water level is larger than the crest level, water will flow
over the dike (m iddle figure). As the water level will accelerate on the innerslope, the flow velocity at the bottom is higher than on the crest. As soon asa critical flow velocity is exceeded, dam age will start to accum ulate (rightfigure). Failure occurs when the total dam age exceeds a certain threshold.

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7

Figure 4
Example ofcalculating damage
due to overflow witha cumulative
overtopping
approach.

4.3

Fragility curves: failure probability given water level
Following these steps for different water levels (and with a schematisedhydrograph shape), gives the failure probability for each of the water levels.
Without further uncertainty, this is 0.0 when the water level causes a total
dam age lower than the critical am ount, or 1.0 when higher. In reality, a lot of
the m odel param eters are uncertain, resulting in a m ore gradual fragility
curve from 0.0 to 1.0 [10]. These uncertainties are explained in the next
chapter.

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8

5

Sources of uncertainty
Last chapter described the m ethod for determining a relation between waterlevel and dike failure (i.e., the fragility curve). In a fully determ inistic setting,there is a specific water level at which the failure probability jum ps from “nofailure” (0.0) to “failure” (1.0). In reality, several uncertainties cau se thisrelation to be m ore diffuse. This chapter described the quantification of these
uncertainties.
The following im age shows the different uncertainties we consider for the
failure m echanism overflow: the dam age (or critical overtopping velocity, as
described in the last section), the water level and the crest height.

Figure 5
Sketch of overflow
with the uncertain
variables indicated.

5.1

Water levels
For each KP-location (shown in Chapter 3), a relation between discharge 𝑄(or flow rate) and water level ℎ is available. This form ula was provided by the
Hokkaido Development Bureau. It was developed based on the results ofrunoff calculations and the water level and flow rate at the observation
stations. We call these rating curves or HQ-relations. The figure below shows
the water levels from these relations, for different discharges, as well as thedesign bank height.

PR3983 • January 2022

9Figure 6
Design bank height
and water level for anumber ofdischarges

Uncertainty in river levels was evaluated based on the relationship between
water levels calculated from the rating curves and water levels observed in
the field in the target river. Observations were collected at Obihiro, TokachiRiver (KP56.73) and Nantaibashi, Satsunai River (KP15.00). The data are
available for five years from 2014 to 2018. Observations were carried out
once a week or during high runoff. The difference between the observed
water level and the water level calculated from H-Q-relation to the observed
flow were gathered. The m ean and variance of these differences were thenused to quantify the uncertainty. This is illustrated for both locations in the
figures below.

WaterLevel - Discharge relation
Tokachi Riv. Obihiro KP56.73

40
39

H.W.L = 38.33 m

Water level [m]

38
37
36
35
Observed data
Q=a(H+b)^2
ND_Average
ND_95% confidence interval

34
33

Figure 7
Calculation of waterlevel dispersion at
Obihiro point(KP56.73)

PR3983 • January 2022

32
31

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000
Discharge [m3/s]

10WaterLevel - Discharge relation
SatsunaiRiv. Nantai Bridge KP15.00

83
82

Water level [m]8180

H.W.L = 79.22 m

79
78
77

Observed data
Q=a(H+b)^2
ND_Average
ND_95% confidence interval

76
75

Figure 8
Calculation of waterlevel dispersion at
Satsunai point(KP15.00)
Table 1 Average andvariance of thedifference between
observed waterlevels, and waterlevels from therating curves.

74
73

0

500

1000

1500

2000

2500

3000

Discharge [m3/s]
Average [m]Variance [m]Tokachi KP56.73-0.1601

0.2937

Satsunai KP15.00-0.05110.2831

Location

When evaluating the uncertainty, the results from Obihiro site were applied
to the Tokachi River and the results from Minam i Obi Bridge site were applied
to all sections of the Satsunai River.

5.2

Bank Height
The high water level set in the river plans for the Tokachi River and theSatsunai River is defined as the "design bank height. The height of the top of
the em bankment is organized longitudinally based on the ground level data
(LP data) obtained from aerial laser surveying, and the m ean value andstandard deviation are calculate d every 0.2 km in the longitudinal direction of
the em bankment to be used as the uncertainty of the embankment height.
The levee height and LP data in the plan were provided by the Hokkaido
Development Bureau.

PR3983 • January 202211

Figure 9
Design bank height
and average bankheight withuncertainty

For m ost locations the uncertainty in the bank height is very sm all. However,for som e locations the bank height is higher than the average bank height,
which also gives a large deviation. In reality, this deviation is skewed towards
positive values (the actual height being larger than the design height), but inour approach a norm al distribution is used. This leads to large probabilities ofa relatively low bank height, resulting in too high failure probabilities. We
advise to fit a different probability distribution to the differences in future
research.

5.2.1

Water level and crest level uncertainties
The relation between water level and discharge (i.e., the rating curve) canchange from event to event. This can be due to bed erosion or
sedim entation, or hydrodynamic effects like hysteresis. Additionally, the crest
level, with a sim ilar effect as water level uncertainty, is not exactly known.
Both uncertainties can combined into a single variable; the uncertainty in“the difference between water level and crest level”.

5.3

Critical flow velocities
In the cum ulative overtopping approach described in Chapter 4, the standard
deviations of the critical flow velocities are given. These can be used toquantify uncertainty on the strength side:

PR3983 • January 2022•

For “plain grass - good cover”: 𝜇 𝑢𝑐 = 1.80 m /s, 𝜎𝑢𝑐 = 0.38 m /s•

For “plain grass - average cover”: 𝜇 𝑢𝑐 = 1.30 m /s, 𝜎𝑢𝑐 = 0.12 m /s

12

•

For “plain grass - poor cover”: 𝜇 𝑢𝑐 = 0.76 m /s, 𝜎𝑢𝑐 = 0.04 m /s

In our approach, the condition of plain grass is "good cover" because thedikes in the Tokachi River are properly m anaged.

5.4

Combining the uncertainties
The different uncertainties (random variables) can be combine d into a single
fragility curve. We used Monte Carlo sim ulations to calculate a fragility curvethat takes into account uncertainty.
Integration into the fragility curve was done by Monte Carlo sim ulation. The
evaluated uncertainties, water level, levee height, and levee slope conditions,were given as random numbers following a norm al distribution with theirrespective m eans and standard deviations. Equation 6 represents theprobability density function of the normal distribution. For the peak waterlevel, the peak flow rate of the target hydrograph was converted to waterlevel using the HQ-relation (Equation 7), and the standard deviation of the
HQ-relations organized in the previous section was used for the water level toderive a random number following a norm al distribution (Equation 8).
Sim ilarly, for the em bankment height, the standard deviation of thedifference between the planned and actual em bankment height was used toderive a random number according to the normal distribution (Equation 9).
⁄
𝑓 (𝑥) = (2𝜋𝜎 2 )−1 2𝑒𝑥𝑝 (−1( 𝑥 − 𝜇 )2 ) ⋯ 6
2𝜎 2

𝑄𝑝𝑒𝑎𝑘
𝐻𝑝𝑒𝑎𝑘 = √−𝑏 ⋯7𝑎ℎ𝑝𝑒𝑎𝑘~𝑁(𝐻𝑝𝑒𝑎𝑘, 𝜎𝐻−𝑄 ) ⋯ 8ℎ𝑏𝑎𝑛𝑘~𝑁(𝐻𝑏𝑎𝑛𝑘, 𝜎𝑏𝑎𝑛𝑘 ) ⋯ 9
If ℎ 𝑝𝑒𝑎𝑘 > ℎ𝑏𝑎𝑛𝑘,. the overflow flow rate is calculated from Equations 2 and 3.
After this, the flow velocity at the em bankment slope is calculated with
Equation 4. Finally, Equation 5 is used to determine if the dam age exceeds
the threshold for levee failure. The state of the slope in Equation 5 also uses
a random num ber following a norm al distribution.
The above operation was repeated 5,000 tim es for each dike segment andwater level, and the probability of breaching was calculated from thefrequency of the number of tim es that the embankment was expected to bebreached (i.e., total dam age exceeding the critical dam age). For illustration,
two of these iterations are shown in the figure below.

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13

n =1

failuren =2

non-failure
Figure 10
Flow of calculating
dike failure
probability using
Monte Carlosimulation.

PR3983 • January 2022

Repeat N times
Probability of failure = nfailure / N

14

6Bank failure probability
In this chapter, we calculate the probability of levee failure using the m ethodof creating fragility curves from Chapter 4, and the described uncertainties in
Chapter 5. First, the levee failure probabilities are calculated at intervals of
about 0.2 km . Next, the breach probabilities for each predefined section areorganized to provide data for the risk assessment of WP3. Finally, the failure
probability of the levee is also calculated for the situation in which failure of
upstream sections is taken into account.

6.1

Calculate failure probabilities
The failure probabilities of levees in the Tokachi River and the Satsunai River
were calculated. The failure probabilities were calculated at intervals of about
0.2 km . As a result, the failure probability of the levees of the Satsunai River
ranges from about 10 -5 to 10 -3. The failure probabilities of the levees of theTokachi River are in the range of 10 -5 to 10 -3, sim ilarly to Satsunai River,
although they varied greatly by location.

Figure 11
Failure probability
per location (KP).
The design bankheight and a highwater level are
shown for reference

6.2

Combine segments to sections
Until now, the failure probabilities where calculated for a levee segment. In
this section we combine the failure probabilities from segments, to sections,to the whole of Obihiro (recall that the definition of segments and sections isPR3983 • January 202215

explained in Chapter 3). When doing so, we want to avoid counting theconsequences of the sam e flood scenario m ultiple tim es. This m eans that a
failure of one segment, reduces the failure probability of a next section,
unless the conditions for which they fail are independent. Com b ining the
failure probabilities m eans that we m ake a realistic choice between combining
probabilities dependent and independent.
•

First of all, we com bine the probabilities conditional to the discharge.
This m eans we assume that a single high discharge causes a highwater level along all flood defenses. This m ight be a little sim plified
for the two rivers (Satsunai and Tokachi), but is probably a very
realistic sim plification, as the high water levels are caused by thesam e rain event..

•

Secondly, segments are grouped into sections based on similarconsequences. Because these sections are close to each other, we
assum e the strength to be dependent. This m eans that the dike will
always fail at the segment with the highest failure probability (forthat discharge).

•

Thirdly, sections are combined per river branch independently (still
conditional to discharge), but we take the order of the sections into
account. Sections are far enough apart for the strength of the dike tobe considered independent. However, when an upstream sections has
failed, we assum e the water level to be reduced such that thedownstream dikes cannot fail anym ore during this event.

To m ake this as specific as possible, we express it with a m athematicalnotation. Com bining failure probability per section:
𝑃𝑓,𝑠𝑒𝑐𝑡𝑖𝑜𝑛|𝑞 = max𝑛𝑖=1(𝑃𝑓,𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑖|𝑞)
In which 𝑃𝑓,𝑠𝑒𝑐𝑡𝑖𝑜𝑛 is the section failure probability, and |𝑞 m eans conditional todischarge 𝑞. 𝑃𝑓,𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑖 is the failure probability of segm ent 𝑖. There are 𝑛segm ents in the section.
Com bine sections into a total failure probability is done in upstream todownstream order (1, 2, 3, …, 𝑛):𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,1|𝑞 = 𝑃𝑓,𝑖𝑛𝑑𝑒𝑝,1|𝑞
𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,2|𝑞 = 𝑃𝑓,𝑖𝑛𝑑𝑒𝑝,2|𝑞 ⋅ [1 − (𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,1|𝑞)]
𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,3|𝑞 = 𝑃𝑓,𝑖𝑛𝑑𝑒𝑝,3|𝑞 ⋅ [1 − (𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,1|𝑞 + 𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,2|𝑞)]
𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,𝑛|𝑞 = 𝑃𝑓,𝑖𝑛𝑑𝑒𝑝,𝑛|𝑞 ⋅ [1 − 𝑠𝑢𝑚𝑛−1(𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,𝑛|𝑞)]𝑖With 𝑃𝑓,𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜,1|𝑞 we express the scenario failure probability, which iscorrected for potential upstream failures. It is called the scenario probability,
because it can be assigned to the consequences of a flood scenario. 𝑃𝑓,𝑖𝑛𝑑𝑒𝑝,1|𝑞
is the independent section failure probability, so without taking upstream
flooding into account.
For the m ost upstream section, the scenario and independent failure
probability are the sam e, as no flooding further upstream is considered. For
the second section, the failure probability is reduced, since failure can only

PR3983 • January 202216

take place when the upstream section has not failed. For the third section,
both upstream sections reduce the flood risk, etcetera.

6.3

Segment probabilities
The result of com bining segment probabilities to sections is shown in the
figure below. Per discharge the m aximum failure probability is selected,
resulting in the black curve. For m ost sections a single segment is dominant,
only for section KP59.6 (lower centre in Figure 12) two sections result in acom bined fragility curve.

Figure 12
Fragility curves forall segments(colored), and the
combined section
curve (black).

6.4

Combining section probabilities per river branch
The result of com bining section probabilities per river branch, is shown in thetwo figures below. The black curves are the original section curves,
independent of upstream sections. The red dashed curves are after taking
potential failure of upstream sections into account. The highest discharges
will certainly lead to upstream failure, reducing the conditional failure
probabilities downstream for these discharges. Therefore, downstream
sections will only fail at relatively low discharges, especially if the section has
a higher failure probability.

PR3983 • January 202217Figure 13
Fragility curves forall sections ofTokachi river, before
and after taking theeffect of upstream
sections into
account.

6.5

Comparing the calculated failure probabilities
To show the sensitivity of the total failure probability to the m ethod withwhich we calculated it, we calculate the total failure probability with fulldependence, full independence and the applied m ethod (m ix of dependence
and independence).

Table 2 Total failure
probability for theapplied method,compared to the fully
dependent andindependentsituation.

Approach

Failure probability

Full dependence

0.00883

Applied m ethod (partial dependence)

0.02564

Full independence

0.07476

The applied m ethod is m ore or less in between the independent anddependent situation. Note that these probabilities are calculated for the“Future” scenario and the “average” discharge statistics. For a different
com bination

PR3983 • January 2022

18

7

Flood probabilities
The m ethod for calculating the flood probability is extensively described inchapter 2. Here, a short overview of the steps leading from discharge
statistics (WP1) to consequences (WP3) is given.
•

WP1 calculates the exceedance probabilities of the discharges for
Satsunai and Tokachi river. Additionally, inform ation on thehydrograph shapes is provided. This is used to define a num ber of•hydrograph classes, with different flood durations.
Fragility curves give a relation between a water level and a failure
probability conditional to that water level. It is therefore a m easure of
the strength of the dike. In WP2 we calculate these curves for the
failure m echanism overtopping. We take into account:othat the duration of overtopping affects the failure probability.

oDifferent hydrograph classes give different fragility curves.
The uncertainty in the revetm ent quality, revetm ent height, andthe water levels. With a Monte Carlo sim ulation these uncertainties
are incorporated in the fragility curve.

This gives a failure probability per dike segm ent, dependent on thehydrograph class. Calculating these is the m ain goal of WP2.
•

WP3 focusses on calculating the flood risk, by com bining the floodprobabilities with the dam age and loss of life it causes. Many of thesections will fail under the sam e conditions. However, if one segment
fails, it is less likely that the others will still fail. We need to consider
this, to avoid overestimating the calculated flood risk. We do this bycorrecting the fragility curves of downstream sections for potential
upstream flooding.

7.1

Discharge statistics
The discharge statistics are determined within work package 1 (WP1). The
results from that work package are used in this work package to determine
the flood risk in WP3. In this chapter a short sum mary is given on the floodprobability. More details on determining flood probabilities can be found in
the report on WP1.

PR3983 • January 2022

19

Figure 14
Discharge statistics
of Past simulation

7.2

Flood probability
The flood probability was calculated using the method explained in Chapter 6,
using the exceedance probabilities of the discharge calculated in WP1. The
flood probability was calculated for each section that is used WP3. It is am easure of the safety of the dike section, and is shown in the m iddle columnof the table below. Because upstream failure reduces the probability of failure
for downstream sections, also the scenario probabilities are calculated inwhich this is taken into account. These are the probabilities that are used in
WP3. The probabilities in the four tables below are for the average discharge
statistics and upper lim it, as well as for the past scenario without climatechange and the future scenario.

Table 3
Past, average
Failure probabilities forthe sections used in
WP3. The scenario
probabilities are thefailure probabilities
after correcting for
upstream failure.

PR3983 • January 2022

Location

Section failure probability

Scenario probability

Satsunai KP7.0

2.53e-05

2.53e-05

Satsunai KP6.4

5.67e-05

3.79e-05

Satsunai KP5.2

4.62e-06

1.45e-07

Satsunai KP4.2

3.00e-07

6.52e-10

Tokachi KP62.4

1.41e-04

1.41e-04

Tokachi KP61.4

3.36e-04

9.93e-07

Tokachi KP59.6

1.21e-03

5.36e-04

Tokachi KP58.0

6.62e-04

4.49e-05

Tokachi KP56.4

2.04e-02

1.06e-02

20

Table 4
Past – upper boundFailure probabilities forthe sections used in
WP3.

Table 5
Future – average
Failure probabilities forthe sections used in
WP3.

Table 6
Future – upper boundFailure probabilities forthe sections used in
WP3.

PR3983 • January 2022

Location

Section failure probability

Scenario probability

Satsunai KP7.0

1.36e-03

1.36e-03

Satsunai KP6.4

2.11e-03

9.30e-04

Satsunai KP5.2

5.42e-04

3.75e-06

Satsunai KP4.2

1.22e-04

1.07e-08

Tokachi KP62.4

6.85e-03

6.85e-03

Tokachi KP61.4

1.36e-02

7.25e-06

Tokachi KP59.6

1.98e-02

2.31e-03

Tokachi KP58.0

1.76e-02

2.64e-04

Tokachi KP56.4

8.52e-02

1.66e-02

Location

Section failure probability

Scenario probability

Satsunai KP7.0

8.87e-04

8.87e-04

Satsunai KP6.4

1.42e-03

6.52e-04

Satsunai KP5.2

3.26e-04

2.75e-06

Satsunai KP4.2

6.11e-05

3.37e-08

Tokachi KP62.4

5.46e-03

5.46e-03

Tokachi KP61.4

9.52e-03

5.72e-06

Tokachi KP59.6

1.55e-02

2.07e-03

Tokachi KP58.0

1.37e-02

2.56e-04

Tokachi KP56.4

7.53e-02

1.60e-02

Location

Section failure probability

Scenario probability

Satsunai KP7.0

1.47e-02

1.47e-02

Satsunai KP6.4

1.99e-02

6.27e-03

Satsunai KP5.2

7.99e-03

2.86e-05

Satsunai KP4.2

3.05e-03

3.49e-07

Tokachi KP62.4

5.30e-02

5.30e-02

Tokachi KP61.4

7.86e-02

1.57e-05

Tokachi KP59.6

9.38e-02

3.46e-03

Tokachi KP58.0

9.16e-02

5.66e-04

Tokachi KP56.4

2.09e-01

1.88e-02

21

8

Sensitivity analysis
The failure probabilities calculated in this study are subject to thequantification of the uncertainties. In this chapter, we first com pare thesensitivity of the param eters discussed in Chapter 5. Additionally, theassum ed hydrograph shape is compared to sim ulated shapes.

8.1

Uncertainty in bank height, crest level and rating
curve
In this study, three factors are considered as random variables. The first is
the relationship between water level and river discharge, the second is theheight of the em bankment, and the third is the condition of the embankment
slope. Here, we conducted a sensitivity analysis to see how m uch the three
uncertainties affect the probability of levee failure.
The m ethod of sensitivity analysis is based on the following conditions:
1.

When all three uncertainties are taken into account2.

when only one item of each is considered (three cases)

3. When no uncertainty is taken into account
The analysis was conducted for the Tokachi River Obihiro site (KP56.7) as arepresentative.
In the case of including the three uncertainties, the failure probability is
distributed between 10,500 m 3/s and 12,000 m 3/s. In the case of including
only H-Q uncertainty, the distribution is alm ost the same range. On the other
hand, in the case of including only Grass Condition and the case of including
only Bank Height, the distribution is in the range of 11150m 3/s to11300m 3/s. This range is narrow and close to the range of cases where
uncertainty is not taken into account. These results indicate that theuncertainty of water level and discharge had a significant effect on theprobability of levee failure at this point.
From the sensitivity analysis, it is possible to analyze the uncertainties that
affect the failure probability. It is also possible that uncertainties with small
im pact m ay not need to be considered.

PR3983 • January 202222

グラフの数字はダミー
1
0
1

2

Included 3 uncertainties
Only included uncertainty in H-Q
Only included uncertainty in Grass Condition
Only included uncertainty in Bank HeightNot included uncertainty

Expansion

Figure 15
Comparison of
Fragility Curves by
Included
Uncertainties

8.2

Hydrograph shape
In this pilot project we assumed a cosine-shaped water level tim e evolution
to calculate the overflow over the dike and resulting revetment dam age. In
this section we discuss the im plications of the assumed shape, by com paring
the cosine shape to a set of hydrographs along Satsunai river. To be specific,
the hydrographs before flooding, in between KP002.80 and KP024.60. In theanalysis, we consider a peak discharge of 4000 m 3/s, and overflow fordischarges higher than 2000 m 3/s. In reality, overflow happens at a certain
water level, but for sim plicity we consider only the discharges, without
converting them to water levels.
The first step is to collect all hydrographs with a peak discharge in between
3500 and 4500 m 3/s, as is shown in the figure below on the left. The range isa trade-off between number of hydrographs and consistency (sim ilar peakdischarge). In the second step, we shift the hydrographs to the same peakm om ent, as is shown on the right in the figure below.

Figure 16
Left: hydrographs
between 3500 and
4500 m3/s, scaled to4000 m3/s. Right:
the samehydrographs, now
also shifted to thesame peak moment.

PR3983 • January 2022

23

We classify the duration of each hydrograph, by considering the duration thatthe discharge is higher than the overflow discharge of 2000 m 3/s. The
histogram of these durations is shown below:
Figure 17
Histogram ofduration above theoverflow discharge
level, for all selectedhydrograph.

Based on these durations, the hydrographs are divided into three bins: thelowest 20% of the overflow durations, the m iddle 60% and the highest 20%.
For each of these classes, we calculate an average hydrograph shape, by
calculating the average discharge at each tim e step within the class. The
result for this is shown in the figure below. In this figure also the used
schematized cosine shape is shown.

Figure 18
Left: hydrographs
between 3500 and
4500 m 3/s, scaled to4000 m 3/s. Right:
the samehydrographs, now
also shifted to thesame peak moment.

The variation in hydrograph shapes between the three classes is not too big,
as we already saw in the histogram . The schematized shape is a good m odel
for the peak of the hydrograph shape, which is the part that m atters most forthe dike safety.
Although we did not explicitly calculate the effect of the uncertainty inoverflow duration, we can assum e, based on this quick analysis, that thehydrograph shape has a relatively sm all effect on the failure probability. For
different locations or river branches, these result m ight differ. Another pointof attention is that the average per tim e step we used, averages out
secondary peaks, as the one shown on the right in Figure 16. In a next
project the sensitivity of the hydrograph shape could b e analysed further,and even be taken into as a random variable.

PR3983 • January 2022

24

8.3

Concluding
Of the sources of uncertainty that were discussed in Chapter 5, theuncertainty in the rating curve has the largest effect on the failure
probability.

PR3983 • January 202225

9Discussion
In this section, the probability of levee failure and flood occurrence isdiscussed in term s of the results of the calculation m ethod approach.

9.1

Failure Probability
According to the calculation results of the probability of levee failure, theprobability of levee failure near KP60.8 of the Tokachi River is sm all, about
10-6. This m ay be due to the fact that this section includes a bridge and isrelatively high com pared to the sections before and after it. On the other
hand, the probability of levee failure near KP58.8 of the Tokachi River isrelatively higher than that of the sections before and after it. Although theaverage height of the em bankment is high due to the inclusion of the bridge,
the difference in the height of the em bankment between the bridge and thesection before and after the bridge is large, and the uncertainty of the heightof the em bankment within the section is thought to have influenced theincrease in the probability of em bankment failure. This point needs to beaccurately evaluated in the future by assessing the uncertainty as only
positive variability.
The results of the uncertainty sensitivity analysis indicate th at the waterlevel-flow relationship has a significant effect on the probability of levee
failure at the target locations. The relationship between water level and flow
is due to the tem poral changes in the riverbed topography. It changes witheach outflow as well as with tim e during the outflow. This result is considered
to represent the flood characteristics of the target watershed, which ischaracterized by high flow velocity and large water level changes.
Three uncertainties are taken into account in the probability of levee failure.
It is unclear how m uch influence this uncertainty has on the risk assessment
in WP3. If the im pact on the outcome of the risk assessment is small, theuncertainty here m ay be negligible. On the other hand, if the im pact is large,
it m ay be necessary to reconfirm the factors that cause uncertainty andconduct sensitivity analysis of the data. In the future, we believe that asensitivity analysis of the uncertainty of the calculation results of each WP to
the risk assessment results should be conducted.

9.2

Flood probability
As for the flood probability, within a section of the sam e river with the sam eplanned size (discharge), the results showed that the levee failure probability
was larger upstream and smaller downstream. For exam ple, in the section of
the Tokachi River after the confluence of the Shikaribetsu River (from KP59.6
to KP56.4), the return period was about 1.5×10 3 years upstream and about
4.5×10 6 years downstream. In the Satsunai River, the return period is about

PR3983 • January 2022

26

3.0 x 10 3 years in the upstream side and about 4.9 x 10 6 years in thedownstream side. This result is due to the fact that the failure probability of
the upstream bank is deducted when calculating the failure probability of thedownstream bank.
The hazard m aps published in Japan show inundation damage based on theplanned size of the river and the assum ed maxim um external force. The
condition for a levee failure is an exceedance of the planned elevation. The
results of the inundation analysis are superimposed on the results of theinundation analysis for all the assum ed breach points. In other words, thelevel of safety of levees is uniform, and levee breaches are considered as
independent phenomena. This m ethod can provide residents with a safer
assessment of the hazard. On the other hand, when quantifying dam age forthe purpose of flood risk assessment, there is a possibility of overestimation.
The approach in this study adopts a non-independent approach that
expresses the probability of levee breaches at each lo cation in a probabilistic
approach and takes into account the occurrence of levee failures upstream.
As a result, this m ethod expresses the probability of flood occurrence in am ore realistic way. In the future, quantitative flood risk assessment will be
required to promote flood control m easures in flood plains. By adopting them ethod proposed in this study, m ore accurate flood risk assessment will bepossible.

PR3983 • January 2022

27

10

Conclusions andRecommendations
The results of this study are described below.
•

A probabilistic assessment of levee failure was conducted for levees in
the Tokachi River basin, targeting the phenomenon of levee failure due
to overtopping. We proposed to incorporate three uncertainties related
to the process of levee failure in the probabilistic assessment. The first is
the relationship between water level and flow rate, the second is therelationship between the planned high water level and the actual leveeheight, and the third is the condition of the levee slope.

•

The probability of levee failure at each evaluation point was calculated
and the results were integrated in several sections. We proposed an
approach to calculate the downstream levee breach probability byconsidering the upstream levee failure probability in the integration.

•

Com bined with the exceedance probability of flow provided by WP1, theflood probability was calculated. As a result, the m axim um and m inimum
flood probabilities were found to be about 1.6×10 3 and 2.4×10 8,
respectively, in the vicinity of the targeted Obihiro urban area.

The approach proposed in this study is expected to contribute to the study offlood control m easures based on quantitative inundation risk assessment in
the target watershed. The future research items proposed in the study are as
follows.
•

In this study, levee failure due to overtopping is the subject ofevaluation. In this study, we focused on levee failure due to water
overtopping, but there are some reports of levee failure due to erosion
and piping in Japan. Therefore, the probability of levee failure due to•other levee failure m echanisms should be evaluated in the future.
In this study, flood hydraulics are calculated assuming a m odel
waveform . Flood runoff in Japan is characterized by various shapes ofhydrographs, which are expected to affect the probability of levee failureand dam age during inundation. Therefore, the flood probability shouldbe evaluated taking into account the shape of the hydrograph.

PR3983 • January 2022

29

11

References[1] Obihiro Development and Construction Department, Hokkaido Regional
Development Bureau, Report of flood in August, 2016. (Japanese Only)[2] Interagency Performance Evaluation Task Force: Performance Evaluationof the New Orleans and Southeast Louisiana Hurricane Protection System –Final Report, Volum e VII, pp.108-109, US Arm y Corps of Engineers, 2007.
[3] Water and Disaster Managem ent Bureau, Ministry of Land, Infrastructure
and Transport, The Guide for Analysis of Flood Dam age Indicators (2013 trial
version), 2013,https://www.m lit.go.jp/river/basic_info/seisaku_hyouka/gaiyou/hyouka/pdf/
higaisihyou_h25.pdf . (Japanese only)[4] Ikeushi, K. et al., Inundation Patterns and Fatality Analysis on Large scale Flood, Journal of Japan Society of Civil Engineers. Ser. B1, Hydraulicengineering, Vol.67, No.3, pp133-144, 2011.
https://www.jstage.jst.go.jp/article/jscejhe/67/3/67_3_133/_article/char/ja/ (Japanese Only)[5] Jonkm an, S.N., Loss of life estim ation in flood risk assessment: Theory
and applications, 2007,https://repository.tudelft.nl/islandora/object/uuid%3Abc4fb945-55ef-4079a606-ac4fa8009426 .
[6] Water and Disaster Managem ent Bureau, Ministry of Land, Infrastructure
and Transport, Technical Review Board for River Em bankm ents Based on the
Dam age Caused by Typhoon No. 19 in 2019, 3rd Meeting, Document 2,
2020,
https://www.m lit.go.jp/river/shinngikai_blog/gijutsu_kentoukai/dai03kai/pdf/
doc2.pdf. (Japanese only)[7] Dean, R. G., Rosati, J. D., Walton, T. L., & Edge, B. L. (2010). Erosional
equivalences of levees: Steady and intermittent wave overtopping. Ocean
Engineering, 37(1), 104-113.
[8] Jongejan, R. B., & Maaskant, B. (2015). Quantifying flood risks in the
Netherlands. Risk Analysis, 35(2), 252-264.
[9] Slom p, R., Knoeff, H., Bizzarri, A., Bottema, M., & de Vries, W. (2016).
Probabilistic flood defence assessment tools. In E3S Web of Conferences (Vol.
7, p. 03015). EDP Sciences.

PR3983 • January 202230

[10] Van der Meer, J. W., Ter Horst, W. L. A., & Van Velzen, E. H. (2009).
Calculation of fragility curves for flood defence assets. Flood Risk
Managem ent: Research and Practice, 567-573.

PR3983 • January 2022

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