In levee system reliability, the length effect is the term given to the phenomenon that the longer the levee, the higher the probability that it will have a weak spot and fail. Quantitatively, it is the ratio of the segment failure probability to the cross-sectional failure probability. The literature is lacking in methods to calculate the length effect in levees, and often over-simplified methods are used. An efficient (but approximate) method, which we refer to as the modified outcrossing (MO) method, was developed for the system reliability model used in Dutch national flood risk analysis and for the provision of levee assessment tools, but it is poorly documented and its accuracy has not been tested. In this paper, we propose a method to calculate the length effect in levees by sampling the joint spatial distribution of the resistance variables using a copula approach, and represented by a Bayesian Network (BN). We use the BN to verify the MO method, which is also described in detail in this paper. We describe how both methods can be used to update failure probabilities of (long) levees using survival observations (i.e., high water levels and no levee failure), which is important because we have such observations in abundance. We compared the methods via a numerical example, and found that the agreement between the segment failure probability estimates was nearly perfect in the prior case, and very good in the posterior case, for segments ranging from 500 m to 6000 m in length. These results provide a strong verification of both methods, either of which provide an attractive alternative to the more simplified approaches often encountered in the literature and in practice.

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Estimates of levee system reliability can conflict with experience and intuition. For example, a very high failure probability may be computed while no evidence of failure has been observed, or a very low failure probability when signs of failure have been detected. This conflict results in skepticism about the computed failure probabilities and an (understandable) unwillingness to make important management decisions based upon them. Bayesian networks (BNs) are useful in these circumstances because they allow us to use observations to improve our reliability estimates quantitatively. @en

Flood risk analysis is necessary to make smart, informed decisions about which risk reduction measures deserve priority. When levee systems play a key role in flood protection, these decisions often translate to which levee improvements should be carried out first. In flood risk analysis, the probability that a levee system fails is a critical component, but one that is wrought with uncertainty. Much research has focused on how to calculate the probability of system failure. However, for levees, what is typically seen in practice is an over-simplification of the system to make calculating the system failure probability easier. @en