Ŕ periodica polytechnica
52/2 (2008) 83–89
web: http:// www.pp.bme.hu/ci
Periodica Polytechnica 2008
Hydraulic failure probability of a dike
Received 2008-03-08, accepted 2008-10-21
This paper is a brief introduction on the determination of
separate ﬂood plain basins, the selection and determination of
characteristic ﬂood stages inducing typical economic impacts,
and the principles of taking the safety factor or the probability
of failure of the ﬂood defences into consideration in ﬂood risk
mapping. The failure probability is the origin from the vari-
ability of the soil physical parameters and from the constantly
changing water level.
Probability of failure ·conventional safety factor ·ﬂood risk ·
dike breach ·soil characteristics ·hydraulic failure
Geotechnical Department, BME, M˝uegyetem rkp. 3. Budapest, H-1521, Hun-
Flood risk mapping is a cartographical representation of ﬂood
and ﬂood damage characteristics of diﬀerent probability. The
maps are basic tools in ﬂood prone areas for land use planning,
for priority setting in the ﬁeld of investments for the establish-
ment or improvement of ﬂood security, and they are also essen-
tial for insurance planning and for increasing the public aware-
ness of risk [14,17,18,20].
The important characteristics of ﬂoods inﬂuencing possible
damages are the expected water level (or the expected depth of
ﬂooding), the frequency or return period of diﬀerent water lev-
els, ﬂow velocity conditions, and ﬂood duration. All of these
characteristics can be represented in a ﬂood risk map.
Flood risk maps are usually compiled for unprotected ﬂood-
plains of river or creek valleys. In such cases the surface of the
water ﬂowing in the river bed can be computed as a variable
unsteady ﬂow in an open channel. Diﬀerent water surfaces cor-
responding to discharges of diﬀerent probability are determined,
and the horizontal projection of the respective water levels to the
terrain indicate the limits of ﬂood of diﬀerent probability. Char-
acteristic depths of ﬂooding are easy to derive from detailed to-
pographic maps or digital terrain models. Such ﬂood risk maps
are usually used for land zoning or for the planning of structural
ﬂood alleviation schemes.
In Hungary, where 97 % of the ﬂood plains are already pro-
tected, we believe that the risk of damages can also be related to
the stability or safety of the ﬂood defence structures, dikes, and
conﬁnement dikes. The length of the Hungarian ﬂood dikes is
more than 4200 km, so the ﬂood risk is primarily a factor of the
stability of the dikes.
2 The inconsistency of soil characteristics
The data or research ﬁndings that support the calculation of
the degree of safety from the parameter of shear strength or the
coeﬃcient of permeability are normally scarce. It is common
practice to calculate the central factor of safety from the aver-
age of research ﬁndings. A designer whose calculation takes
into account the smallest of the available measurement results
against the most unfavourable combination of loads exercises
Hydraulic failure probability of a dike cross section 832008 52 2
utmost caution. This calculation yields lower resistance values
than the degree of safety calculated from averages. If a system
still complied with the required degree of safety, the designed
structure must have been uneconomically large. That has led
on to a paradoxical situation: as spending on exploration grew,
more and more studies were performed and the likelihood of
receiving poorer and poorer resistance values kept increasing
along with the safety of designing, which in turn kept driving
the cost of construction higher.
Several researchers have studied soil characteristics as statis-
tical values, such as the distribution and variability factor of soil
features (Table 1). However, a review of the literature failed to
identify data concerning studies on the coeﬃcient of permeabil-
ity and the type of distribution.
As a material used for supporting loads and for construction,
soil is a substance that exhibits utmost variation in homogeneity.
While a ten percent coeﬃcient of variation (Cv=10%) repre-
sents poor quality for concrete, the value of Cv=0.4 should be
viewed as satisfactory with some soil characteristics (see Fig. 1).
Fig. 2 shows the results of 54 studies concerning the angle
of internal friction and 91 studies of shear strength of the ex-
plored sandy and rich clay soils in the ﬂood area of Köröszug.
The results clearly demonstrate the relatively low coeﬃcient of
variation for sand.
3 Safety of ﬂood dikes
The ﬂoods after 1945 have caused 140 embankment failures,
of which 83 (58%) were due to overtopping (52 during the 1956
icejam ﬂood on the Danube), 23 (16 %) to hydraulic soil failure,
10 (7 %) to saturation and 2 (1,5%) to leakage along structures,
other identiﬁed 11 (7,5%), while no cause could be identiﬁed
positively in the case of 14 (10%) [15, 16, 18, 19]. In the pro-
tected ﬂood plain basins the occurrence of the various loss types
can be related to the ﬂood stages aﬀecting the stability or safety
of the ﬂood defences. The total obtained is 143 instead of 140
due to the fact that in three cases diﬀerent mechanisms of failure
were named, which could not be judged as to their correctness.
Evidently, the completeness of the list cannot be guaranteed.
Improvements over the past 150 years involved but rarely any
change in the original trace of the embankments. Explorations
of the subsoil and soil mechanical tests have been introduced as
late as 3540 years ago, which recently revealed that the original
trace passes over areas with adverse soil conditions, where the
soil proﬁle contains:
– the meander crossings with its diﬀerent soil layers,
– layers of organic soil or peat,
– dispersive soils,
– loose, poorly graded ﬁne sands in the vicinity of the surface,
The programme for the investigation of 4200 km ﬂood dikes
was compiled in the 1980s for exploring the subsoil of ﬂood
embankments and for identifying the potential sections of piping
failure. The basic considerations underlying the method are as
– the subsoil under long embankments of moderate height must
– the soil proﬁle must be explored continuously (virtually by
– the subsoil consists generally of a cohesive cover over layers
becoming increasingly coarser with depth.
15 16 17 18 19 20 21 22 25 26
Heavy clay's inner friction angle
Number of samples
Fig. 2. The angle of internal friction of soils explored in the ﬂood area of
In order to carry out the investigation on the stability of the
dikes, the study must be divided into characteristic sections,
within which the following should be presumed more or less
– the high of the crest,
– the stratiﬁcation of foundation soil and the quality of the lay-
– material of the existing dike as well as that of the reinforce-
ment or new defences,
– typical cross-section of the existing dikes, and
– phenomena observed along the dikes during ﬂoods.
The section conforming to the characteristics of the founda-
tion soil has a special importance and needs special care. In
the course of the investigation the safety of the embankments
Per. Pol. Civil Eng.84 László Nagy
Tab. 1. The distribution and variability factor of soil features
Soil properties Distribution Coefﬁcient of variation
Normal Lognormal Other
66 % Corotis  33 % Corotis  Pearson IV or VII 0,15-0,19 Rétháti 
Davidson  Rétháti  0,02-0,2 Borus-Rév 
Brust  0,011-0,028 Borus-Rév 
Ike  0,03-0,05 Evangelista 
Particle density Shultze (1971)
Void ratio 80 % Shultze 
Saturation Rétháti 
80 % Shultze  33 % Corotis  Rétháti  0,11-0,38 Rétháti 
66 % Corotis 
Plasticity limit Lumb  Corotis  0,04-0,10 Borus-Rév 
Plasticity index Lumb  Rétháti  0,26-0,54 Rétháti 
Hooper  0,15-0,31 Morse 
Insley  0,17 Weber 
Wu  0,05-0,14 Schultze 
Friction angle 50 % Shultze  0,06-0,11 Harr 
Cohesion Lumb  0,42 Weber 
0,26-0,68 Lumb 
Fig. 1. Coeﬃcient of variation of soil characteris-
concrete water content wet density folyási határ shear test cohesion friction angle perme. coef.
Coefficient of variation
against piping failure is determined by successive approxima-
tions involving several disciplines, like geophysics, hydraulics,
soil mechanics, and surveying. To determine the longitudinal
proﬁle of the long dikes and the individual sections, one of the
best methods is the permanent horizontal geo-electric probing
with a 1,0 meter electrode distance. The application of this
method makes the exploration of continuous stratiﬁcation pos-
sible. This method also reduces the cost of exploration, while
the application of more expensive methods may be required less
often and only for the identiﬁcation of the layers at easily deter-
4 Determination of the conventional safety factor
Controlling the safety factor of the embankment divided into
characteristic sections must be accomplished section by section,
according to standard methods speciﬁed in appropriate guide-
lines and standards. The conventional safety factor is:
where Ris the resistance (or strength), and Qis the action ef-
fect (load). Using and transforming the equations determine the
safety factor of the defences at actual water stages, and the ﬂood
levels corresponding to previously selected safety factors can be
Hydraulic failure probability of a dike cross section 852008 52 2
Fig. 3. Failure probability at diﬀerent water stages
Fig. 4. Better and worse dike
Per. Pol. Civil Eng.86 László Nagy
Fig. 5. The occurrence probability of a failure of a dike
So we have the opportunity of deﬁning the ﬂood hydrograph
peaking at the level corresponding to the loading capacity of the
defence structure. Since the most vulnerable cross sections of
the defences are also known, the ﬂood hydrographs represent-
ing the loading capacity are to be transformed to these possible
breach points. The loading capacity of the defences can be deter-
mined by repeating the computations carried out earlier in order
to deﬁne the extension of the ﬂoodplain of 1 % probability of
inundation, the extension of the ﬂood plain section threatened
by the stage corresponding to .
Advanced dimensioning methods consider both the impacts
inducing (Q) or hindering (R) the breach to be independent and
probabilistic variables. It is obvious that from the viewpoint of
stability all the combinations of load and resistance are disad-
vantageous where R<Q, represented in the ﬁgure with the
barred territory. The size of this territory is equal with the fail-
ure probability and therefore is appropriate for characterizing
the magnitude of risk of the given section.
5 The probability of failure at ﬂood dikes
In ﬂood protection dikes both load and resistance develop
along certain probabilities. Load is interpreted in terms of the
probability of water levels. The variation of soils and soil char-
acteristics prevents us from identifying in other than probabilis-
tic terms what resistance to failure a ﬂood protection dike will
have under certain water level loads (probable water levels).
When calculating the probability of failure, Q(w) is used to rep-
resent the load probability function, as it is the function of water
levels, whilst R (w) stands for the probability of resistance func-
tion, as it has been calculated from water levels.
The relation between load and resistance may be expressed
by the safety margin (SM):
SM =R(w) −Q(w), (2)
which is also a probabilistic variable. The failure probability
expresses the probability of the opportunity of load exceeding
The failure probability can be determined either from the avail-
able soil physical data, applying probabilistic design methods
for the whole calculation system or from the traditionally cal-
culated safety factors using a semi-deterministic approach. For
ﬂood dikes the value of failure probability generally must be:
Hydraulic failure probability of a dike cross section 872008 52 2
Fig. 6. The hydraulic failure probability of an old
and a developed dike
0.0001 0.0010 0.0100 0.1000
water st age
= 0,41% P
which means that among all possible combinations of load and
resistance values only 1 % would lead to breach. In other words,
in 1 % of possible cases will be Q(w) > R(w).
I have prepared a detailed calculation to evaluate the safety
of the protected ﬂood areas along the Upper Tisza and the Sajó
rivers. Based on the calculations and the dike failures of the past
35 years, it is recommended to provide the
probability of failure of a cross section. At the present stage
of the research, it can be identiﬁed as a boundary value for the
probability of inundation (the probability of failure of a ﬂood
control dike multiplied by the probability of a ﬂood event)
The calculations suggest that these values may also be applied in
safety mapping. At present, there is no requirement in Hungary
that speciﬁes an acceptable value for the probability of failure.
It would only be proper to ask why would we use failure prob-
ability instead of the safety factor that we became accustomed
to in practice? The answer is:
•we can characterize the system of defence structures,
•we can obtain the reliability of our results (uncertainties can
be handled), and
•evaluation of risk is possible.
The hydraulic failure probability of a dike with conventional
geotechnical methods can be caluclated for a given water stage.
Repeating the calculation for more water stages gives the fail-
ure probability as a function of the height. Fig. 3 represents the
results of the calculated values of the failure probability in the
possible range of water stages, in addition to the probability of
occurrence of water stages in case of a given proﬁle of a dike
. Diﬀerent failure probabilities are depicted on Fig. 4. de-
pending on the water level. Since the failure probability and
the occurrence of water stages are independent, the probabil-
ity of their joint occurrence can be calculated as the product of
the multiplication of their probability, that is R(water level) ·Q
Naturally, we are only aware of the size of resistance (R) and
size of load (Q) functions to a certain level of probability as both
are probability variables (Fig. 4).
Investigating the R(w) ·Q(w) function, the occurrence prob-
ability of a failure of a dike proﬁle can be characterised by the
maximum value of R(water level) ·Q(water level) function.
This consideration is interesting enough for further investiga-
The hydraulic failure probability of a dike at a certain water
stage is shown in Fig. 6. After the proposed development the
new dike failure probability is less then twenty % of the old one.
How safe is any given dike? The answer is provided by a
probabilistic risk assessment, the beneﬁts of which were de-
scribed along with a standard for tolerable risk. It was stressed
that in the absence of analytical techniques, the diﬃculty of as-
signing probabilities can be addressed through the use of experi-
enced engineering judgement who is familiar with the dike and
with all investigations and previous studies at their disposal. It
was proposed that a risk could become a systematic and com-
prehensive framework for the application of engineering judge-
Risk is the product of failure probability and consequences of
the failure. The application of failure probability in the evalua-
tion of existing and also in design of new ﬂood defence struc-
tures gives us the possibility of adapting these problems to the
risk standards. A standard for tolerable risk is needed in con-
junction with a risk analysis to evaluate dam safety, its purpose
being to permit decisions on dike safety remedial work to be
based directly on risk in a consistent and quantiﬁable manner.
Per. Pol. Civil Eng.88 László Nagy
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