Probabilistic Aspects of the Seepage Flow in Dikes

 

R. Pohl

 

Dresden University of Technology,

Institute for Hydraulic Engineering and Applied Hydromechanics

Priv.-Doz. Dr.-Ing. habil. R. Pohl

Institut für Wasserbau und THM

Technische Universität Dresden, 01062 Dresden (Germany)

Tel: 0049-351-463 5693, fax: 0049-351-463 7141

e-mail: pohl @bbbrs5.bau.tu-dresden.de

 

 

Abstract

In dikes and aquifers where the internal soil composition is not completely known uncertainties can arise when determining the soil properties. By means of a statistical model an attempt is made to find more evident and comprehensive results concerning the flow pattern and the stability of dikes.

 

Keywords: Dike, Levee, Seepage, Permeability, Slope Stability, Probabilistic Hydraulic Design, Statistical Trial

 

The at Present Mostly Used Design

The designs in hydraulic engineering and in other disciplines of civil engineering according to generally accepted rules of technology are mostly intended for design in specific loading cases which are assumed to represent the decisive case (Figure 1).

 

 

Figure 1: At present mostly used practice of design in hydraulic engineering

 

The decision on the "decisive" loading case is rendered more difficult, however, by different kinds of uncertainties. In hydraulic engineering the following uncertainties can be expected: The natural uncertainty results from the randomness of the courses of meteorological and hydrologic processes, and it constitutes the greatest problem, which cannot be eliminated objectively. Possible measurement or transmission errors may lead to a data uncertainty. For purposes of design the reality is modelled. Since complete transmissibility is not possible in many cases, since the hydraulic model laws cannot be fulfilled at the same time (Froude, Reynolds), and since for the formulation of mathematical relationships approximation functions were used in many cases, there can be expected a model uncertainty. The parameter uncertainty results from the fact that possibly not all dependences or parameters are known or determinable. Even if these uncertainties entering design did not exist, there may occur operational uncertainties during the service life of the structure (operating errors, manoeuvring errors, fluctuations of demand, failure of components).

 

Proposal for Design on a Probabilistic Basis

Because of the mentioned uncertainties, sometimes there cannot be formulated definite loading cases. In such cases, the engineer occasionally uses the remedy of estimating upper and lower limiting cases (on the "safe" and "uncertain" side) But even this extension of design gives an insight into the possible situations of loading in points only here and there.

It would be advantageous to carry out the design not only with two values but with a large number of values whose frequency of occurrence corresponds to the distribution observed in reality. If the measured values are erroneous, these may scatter, e. g., normally distributed around an expected value. If there are natural influencing quantities (flood, wind), inter alia the Gumbel distribution, the Pearson (III) distribution or the lognormal distribution come into consideration.

 

If one succeeds in contrasting one random loading quantity (characteristic of distribution) with one random resistance, a closed-form (exact) mathematical solution of the so-called convolution integral is possible when certain conditions are met [3], [5]. If there are several random input quantities the mathematical expenditure increases considerably. Then, in many cases a closed-form solution can be given no more or only with considerable simplifications. In those cases, the distribution function of the required input quantity can only be determined with the aid of statistical trials (Monte Carlo method) [6].

The following proposal can be formulated for the design (Figure 2):

 

distribution of input data	transformation formula or procedure	distribution of results
	(process function) 1. transformation of input value(s), exact solution (e.g. wave run-up, overflow)   2. in more difficult cases n repetitions of transformation with random numbers (statistical testing) ß-------	Q

 

Figure 2: A proposal for design in hydraulic engineering

 

The Situation in the Hydraulic and Static Design of Dikes

Whereas for the construction of new dikes the soil materials used are well known with respect to their hydraulic and soil-mechanical properties and can be also kept within a very small range of scatter, in historically grown dikes only explored in places the soil properties mostly are only incompletely known. Interpolating is necessary between the points of exploration. Results of sampling are subject to a variance, particularly since undisturbed sampling is difficult.

Considerable inhomogeneities and changing soil strata are found, e. g., also on the river Oder in the historically developed Brandenburg dikes, whose height was increased several times. Adjacent soils partly have very different permeabilities.

Figure 3 is based on a dike cross-section near Reitwein (km 5.1). Clearly discernible is the silt body on the inner side face, which very probably corresponds to the original dike, whose height was later increased with silty sand.

By the calculation of the seepage flow (as a 2-D vertically plane problem) it was found that the seepage flow line considerably differs from that in a homogeneous dam with the same geometry because of great inhomogeneities in the dike body and in the subsoil. Since the dikes do not have seepage prisms, the seepage flow line frequently comes out on the air-side face of the dikes, sometimes already on the berm, so that (retrogressive) erosion may occur. If the origi­nal dike is less permeable than the cover, the saturated zone is raised (Figure 3a). In the case of Figure 3c the more permeable older dike has the effect of a drainage prism.

 

a
b
c

soil		kf (m/s)	g (kN/m³)	g' (kN/m³)	c (kN/m²)
	OH	10-7	11	1	1
	SU	5. 10-6	19	10	1
	SU*	10-6	20	10	3
	TL	10-9	19	9	3
	UM	10-8	18,5	8,5	4
	SE	4. 10-4	19	10	-

 

 

Figure 3: Dike cross-section (similar to the Oder dike near Reitwein, km 5.19

a: explored, simplified cross-section, old dike less permeable than cover

b: without covered dike, c: covered dike with greater permeability, acting as drainage

 

Such a deviation of the flow pattern from that in the homogeneous dam has also been found in many other dike cross-sections in the region.

If the inner side face of dikes or dams is covered with a layer of low permeability, the seepage flow line is elevated. This was observed, e. g., at the canal embankment of the Oder-Spree Navigation Canal near Eisenhüttenstadt during the Oder flood in 1997 in the form of moisture depositions and confined water under the covering layer. If there are cover layers of low permeability on the waterside face, the seepage flow line may be lowered, water continuing to move within the unsaturated zone above the seepage flow line (cp. Figure 4).

 

Figure 4: Probabilities of falling below for the position of the seepage flow line due to data uncertainty of permeabilities (schematic representation for the canal embankment of the Oder-Spree Canal near Eisenhüttenstadt)

 

An Example of the Design of Dikes on a Probabilistic Basis

If the interior of the dam is not sufficiently explored, deviations may occur between the calculated seepage flow field and the observations concerning moisture depositions and outflow of water.

There cannot be achieved a qualification of the result, however, when the data uncertainty which is connected with the sampling is recorded by probability distributions for the soil properties within possible fluctuation ranges. By calculations repeated n times with random combinations of the soil properties there are obtained n results with respect to the position of the seepage flow line or stability, which in turn could be described by a distribution. This method of statistical trials (Monte Carlo Method - MCM) yields seepage flow lines that will be fallen below with a certain probability. This is schematically represented in Figure 4. It was assumed that the permeability coefficients for the present soils may deviate upward and downward by up to one power of ten and have a lognormal distribution in this range.

 

 

Figure 5: Stability with 95 % (top) and 20 % (bottom) probability of exceeding according to Figure 4

 

The results of calculation for stability show that for an unfavourable case (for which it is expected with a probability of 95 % that stability is greater) a slope failure is expected because of h = 0.9 (Figure 5 top). At a corresponding combination of the permeability coefficients there may also result a sufficient stability as in figure 5 below with h = 1.53 (only with a probability of 20 % an even higher stability is expected). Thus there are two different statements for one cross-section.

 

Figure 6: Probability of exceeding for the safety against sliding failure of the slope

 

If the probability distribution on which Figure 4 is based is used for calculating the stability of the inner slope of the dike, there can be given a probability distribution of the safeties achieved in each case as against slope failure by sliding (Figure 6). Consequently, the probability of failure with the given geometry and the given water level in the steady seepage-flow case is about 8 % (failure criterion h < 1). With a probability of 55 % the safety exceeds the value of h = 1.3 which is aimed at. There is to be taken into account that the probabilities are conditional ones, which presuppose the occurrence of the design water level and are only in­fluenced by the data uncertainty with respect to the permeability coefficients.

 

The photograph in Figure 7 shows that the case of failure calculated with P = 8 % unfortunately has occurred. A factor may also have been the live loads by driving on the dam crest in connection with the defending of the dike, which were not taken into account in the calculation. It can be seen that the slid­ing plane begins on the crest as in the calculation (Figure 5).

 

 

Figure 7: Canal embankment of the Oder-Spree Canal, Eisenhüttenstadt: slides on the air-side face (photograph: author)

 

Reliable Probabilities of Failure

The interpretation of this value requires the specification of reliable probabilities of failure. So far, however, there have not been available any generally recognized values. In the literature there are only proposals by different authors for the respective structures, which depend on, inter alia, the construction and the potential danger.

Freudenthal [3] proposes P £ 10^-2 for the permissible total probability of failure of a large structure if the cost of damage is not considerably higher than the cost of reconstruction. If the cost of damage is much greater, he recommends P £ 10^-3, i. e. one case of failure in 1000 years on average. For the overflowing of sea dikes due to flood in the Netherlands there is permitted P £ 10^-4 in more densely populated regions and P £ 2.5·10^-4 in less populated regions.

The total failure in the example stated above is between 10^-2 and 10^-3, allowing for the flood probability in the range of the values given by [3].

 

Conclusions, Outlook

With the aid of a statistical model it has been attempted to qualitatively describe the collective of possible results of calculation, which may be obtained due to the scatter of the input data, with a probability distribution. Including the probability of occurrence for the flood level used, in the example of calculation presented there results a probability of failure for the stability of the air-side slope between 10^-2 and 10^-3 per year in dependence on the recurrence interval of the flood level. For the practical application of the described method of design acknowledged values for permissible probabilities of failure will be necessary in future.

Although in public a contradiction in acceptance exists between voluntarily assumed risks (using of means of transport, risky kinds of sport, life insurance) and involuntarily assumed risks (industrial plants, potential danger from certain engineering structures), it can be assumed that the probabilistically determined way of thinking and corresponding design methods will continue to prevail.

Probabilistic Aspects of the Seepage Flow in Dikes

 

References

[1]    Buß, J.: Unterströmung von Deichen.- In: Mitt d. Franzius-Inst. der TU Braunschweig 1987, (92) (in German)

[2]    Franke, D., Pohl, R., Engel, H., Niesche, H. Krüger, F.: Das Oder-Hochwasser 1997, Ursachen der Deichschäden.- In: Wasserwirtschaft-Wassertechnik 1998 (in German, )

[3]    Freudenthal, A.M.: Safety and reliability of large engineering structures.- In: Symposium of the National Academy of Engineering, Washington, D.C. 1970

[4]    Ludewig, M. ; Pohl, R.: Stauanlagen (Kap.4).- In: Wiegleb, K.: Taschenbuch Verkehrs- und Tiefbau, Bd. 4 .- Berlin: Verlag für Bauwesen, 1990, ISBN 3-345-00297-3 (in German)

[5]    Plate, E.: Statistik und angewandte Wahrscheinlichkeitslehre für Bauingenieure.-Berlin: Verl. W. Ernst & Sohn, 1993 (in German)

[6]    Pohl, R.: Die Überflutungssicherheit von Talsperren.- In: Dresdner Wasserbauliche Mitteilungen 11/1997, TU Dresden, Institut für Wasserbau und Technische Hydromechanik ISSN 0949-5061, ISBN 3-86005-186-5 (in German)