Reliability based design of flood defenses and river dikes

In this paper an example is elaborated of the design of a river dike, incorporating reliability-based models. The design is demonstrated with the help of this example, which is based on [1]. 1. PROBLEM DESCRIPTION A river dike has to be designed to protect a particular region of land. The goal is to minimize the construction cost and the expected loss, due to inundation. The total cost can be expressed as: Ctot = Cconstr + E(S) ( 1 ) where: Cconstr = dike construction cost E(S) = present value of the expected loss The crest height, h, and the slope angle, α, (assumed to be the same for the outer and the inner taluses) are taken as primary optimization variables. The cost of inspection, maintenance and repair of the dike are not taken into consideration. The interest rate and the service life of the dike, however, are included. In figure 1 a sketch is given of the dike to be designed and of the region to be protected. Figure 1. Plan of river and polder and cross section River It is assumed that one single flood wave raising the water level in the river once throughout a year poses the only hazard to the dike. The shape of this flood wave in time, presented in figure 2, is parabolic. The river width, B, is assumed to be constant, being 400 m. The bottom level of the river, hh , is 3.5 m below datum level. The gradient of the river bed, I, is 10, whereas De Chezy’s constant is taken as C = 40 m s. Figure 2. Shape of the flood wave and indication of the water level River dike The length of the river dike, Ld , is 20 km. Its schematized cross section (not to scale) is shown in figure 3. The cross section is symmetrical in shape. The dike consists of a sand core, the outer talus of which is covered with clay. The crest is at a level h0 above datum level, the crest width is bk and the base width is L2. The subsoil consists of sand, covered with a clay layer thick dk = 3.5 m. The taluses slope with angles α with the horizontal. If necessary a foreland can be created on the outer side of the dike (i.e. river ward) by building the dike further inland. The width of the foreland then is L1. Figure 3. Cross section of the river dike Protected region The protected area, A, covers 200 km. Its level, hm , is supposed to be entirely at 0.5 m above level datum. Part α1 of the region is urban area (α1A km), α2A km is used for agriculture and the α3A km remainder, is industrial area. The coefficients, α1 , α2 and α3 are taken as 0.06, 0.93 and 0.01 respectively. The region is assumed to contain no flood water retaining provisions, neither intended nor accidentally present. Considered failure mechanisms Failure of flood defences can be adequately modelled with help of fault trees (Vrijling, 2001). In this paper, it is assumed that inundation of the protected area can be caused in two ways: a) as a result of overtopping of the dike, not causing collapse; b) as a result of a dike breach, for which the following mechanisms are considered (In practice, more mechanisms need to be investigated.): 1) macro-instability of the inner talus 2) piping (using Lane’s criterion [2]) 3) micro-instability of the inner slope (The fact that in general this mechanism is irrelevant in Dutch circumstances does not exclude it from possible mechanisms). Figure 3: Subtree from total fault tree of river dike failure 2. STOCHASTIC VARIABLES In table 1 the variables considered in the example in hand are given. The majority of these variables are taken as deterministic, mainly for simplicity sake. The following variables are supposed to be stochastic: a. Flood wave • yearly maximum daily averaged water level, ĥ : The yearly maxima of the daily averaged water level, ĥ, are supposed to be exponentially distributed (extreme type III for maxima, or Weibull distributed, with parameter k = 1, see [1]). Adopting μĥ = 3 m and σĥ = 0.9 m, the (cumulative) distribution of ĥ is: 2.1 0.9 ˆ ( ) 1 h F e ζ ζ − − = − ( 2 ) • duration T of the high water level, associated with the flood wave. The duration T is assumed to be log-normally distributed. A research of the discharge data of the river Rhine [1] showed that a lognormal distribution could satisfactorily be fitted to T. The adopted probability density function is: 2 (ln( ) ) 2 1 ( ) 2 y y T y f e τ μ σ τ τ σ π − = ⋅ ⋅ ⋅ ( 3 ) where: y = ln ( T ); y is normally distributed μT = mean of T, taken as 7.5 days σT = standard deviation of T, taken as 4.5 days. b. Soil The following parameters of clay and sand are assumed to be stochastic: k = permeability [m/s] φ = angle of natural slope []; c’ = cohesion [kN/m] λeq = kk,eq / dk,eq = equivalent leakage factor of the clay layer on the outer talus, including the perforations in this layer kk,eq = equivalent permeability of clay dk,eq = equivalent thickness of the clay layer (see below) In an actual case it is sometimes possible to provide a statistical basis for some variables, but for other variables it will be necessary to rely on estimates. c. Geometry • Thickness of clay layer on outer talus. The permeability of the clay layer on the outer talus is important as it is a determining factor in the position of the phreatic surface. The effect of a variation in thickness is combined with the permeability of the clay to the equivalent leakage factor: λeq= kk,eq / dk,eq ( 4 ) • Thickness of clay layer under the dike. The clay layer on which the dike is situated, varies in thickness. It is modelled by a layer of constant but unknown thickness, which is normally distributed with mean 3.5 m and coefficient of variation of 0.2. d. Model factor for piping In the piping mechanism a model factor is introduced as (among other things) to represent the variation in the results. Lane’s formula [2] is chosen to model piping (being a “conservative” assumption compared with Sellmeijer’s [3] formulation). The model factor is supposed to be a normally distributed variable with mean 1.67 and coefficient of variation 0.2. e. Width of the breach The width of the breach in a dike, b , may vary greatly [4]. The distribution is assumed to be of lognormal type. The mean width is taken as 100 m and the coefficient of variation as 1.0. Table 1 Overview of the problem variables variable description type μ unit σ / μ ĥ highest water level (upstream) E 3 m 0.30 T duration of high water level LN 7.5 days 0.60 ck cohesion (clay) N 10 kN/m 2 0.20 φk angle of repose (clay) N 20 degrees 0.20 kk permeability (clay) LN 10 -8 m/s 1.60 cz cohesion (sand) D 0 kN/m 2 φz angle of respose (sand) N 35 degrees 0.10 kz permeability (sand) LN 10 -5 m/s 0.50 dk thickness of clay layer under dike N 3.5 m 0.20 λeq equivalent leakage factor of clay layer on the outer talus (= kk,eq / dk,eq) LN 210 -7 s -1 1.00 b width of the breach LN 100 m 1.00 m model factor (piping) N 1.67 0.20 Ld length of dike D 20 km A protected area (polder) D 200 km hm surface level D 0.5 m α1 fraction of region that is urban area D 0.06 α2 fraction of region that is agricultural area D 0.93 α3 fraction of region that is industrial area D 0.01 s1 max. loss urban area per unit area D 400 NLG/m 2 s2 max. loss agricultural area per unit area D 1 NLG/m 2 s3 max. loss industrial area per unit area D 200 NLG/m 2 r’ actual interest rate D 0.02 fb construction cost per unit volume D 10 NLG/m 3 h0 crest level (above datum level) V 6 á 10 m tan(α) tangent of angles of taluses V 1:2.5 to 1:5 bk crest width D 3 m ρk density of clay (dry / wet) D 1,400/1,900 kg/m ρz density of sand (dry / wet) D 1,600/2,000 kg/m ρw density of water D 1000 kg/m B river width D 400 m hb distance from datum level to river bed level D 3.5 m bs bottom width of ditch D 1.0 m hs distance from datum level to ditch bottom level D 0.5 m C De Chézy’s constant D 40 m/s Ib gradient of river bed D 10 -4 g acceleration due to gravity D 10 m/s L1 width of foreland D/V 0 / var m L2 width of base width of dike D var m n porosity (pore ratio) of sand D 0.4 Cs Lane’s constant D 6 D = deterministic LN = log-normal E = exponential (extreme III, k=1) N = normal V = variable (design value) 3. CALCULATION PROCEDURE 3.1. Optimization of the dike stages As stated in section 1, the total cost, Ctot, has to be optimized. To do this, an appropriate choice of the design parameters, i.e. the crest height, h0 , and the tangent of the taluses, tan(α), must be made in order to establish the minimum of equation ( 1 ). In the calculation procedure the minimum was achieved by assigning a limited number of discrete values to h0 and tan(α), namely = 6, 7, 8, 9 and 10 m, and tan(α) = 1:2.5, 1:3, 1:3.5, 1:4 and 1:5. 3.2. Construction cost The cost of the dike is assumed to depend on the volume of the dike body only. The equation expressing the construction cost is: Cconstr = Ld . h0 . (h0 . cotan(α) + bk ) . fb ( 5 ) in which: Ld = length of dike h0 = crest height (above datum level), design parameter cotan(α) = cotangent of the slope of the inner and outer talus, design parameter bk = crest width fb = construction cost per unit volume. 3.3. Present value of the risk Inundation will occur upon failure of the dike. The result will be a particular amount of damage or loss, S, which is assumed to depend only on the inundation depth, d. The risk in year i, or the expectation of the loss in year i is: E(S)i = P(Fi) . S ( 6 ) where: P(Fi ) = probability of failure in year i S = amount of damage or loss. Considering the actual interest rate and the intended service life of the dike, the capitalized loss expectation, or the present value of the risk, can be written as: , 1 ( ) ( ) (1 ) N i i i P F S E S r = ⋅ = + ∑ ( 7 ) in which: N = intended service life r’ = actual interest rate. If N is large and P(Fi ) is constant in time, E( S) can be written as: , ( ) ( ) i P F S E S r ⋅ = ( 8 )