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Alternative model for piping prediction

The Shields–Darcy (SD) model is a piping model which predicts internal erosion below dikes and the allowable gradient across a dike during high water. In such situations, groundwater flows through sand layers from the river to the polder. This makes the hinterland wet. With a rising water level, the groundwater can also transport sand. If the high water persists for a sufficiently long period, the dikes might eventually fail due to the piping failure mechanism.

The SD model calculates various groundwater discharges and pipe characteristics, such as the critical values of the pipe velocity and pipe discharge. In this piping model, the allowable gradient is determined by two resistors, namely the seepage resistance and the pipe resistance. The seepage resistance refers to the resistance experienced by the groundwater in the sand layer, and the pipe resistance represents the flow resistance in the meandering and braided pipes.

Sand boils (Rijkswaterstaat)
Schematisation of resistors below a clayey dike (application of Ohm’s Law)
Schematisation of hydraulic gradients

From the river, groundwater flows not only to the pipes but also via the sand layer and far to the hinterland. If a control volume is considered underneath the dike, then applying the continuity equation shows that the amount of water from the pipes equals the difference between the groundwater that is entering and leaving. In the SD model, the allowable gradient is schematised by two straight curves, namely the critical pipe gradient (Shields term) and the critical sand gradient (Darcy gradient).

Various hypotheses are available for calculating the allowable gradient. In the SD model, the theories of Ohm, Hagen–Poiseuille, Darcy–Weisbach, Shields, and Grass have been applied, resulting in a set of equations. The unknowns herein have been calibrated and validated with experiments on a small, medium and large scale. Because the differences between calibration and validation are minimal, the study has concluded that the calibration is correct. In addition, the unknown parameters have been successfully verified independently of these tests.

Recently, Hoffmans Advice compared the SD model with the current piping model as proposed by Sellmeijer. Although the SD model has greater predictive power compared to Sellmeijer (95% versus 77%), the differences between tests on a small, medium, and large scale are small. However, for prototype conditions, large differences in the required seepage length occur, especially in the area where the hydraulic conductivity varies from 10⁻⁴ to 10⁻⁵ m/sec. These differences can amount to a factor of 2; that is, the required seepage length according to the SD model can be half that of the current piping model.

The SD model is physically correct and should be further validated in relation to Sellmeijer II, especially for practical situations. Validating both piping models leads to greater substantiated water safety with a potentially substantial cost reduction for the Flood Protection Programme (in Dutch: Hoogwaterbeschermingsprogramma [HWBP]) in the order of several hundred million euros, given the future task of reinforcement. The cost analysis should include measures on the macro-stability failure mechanism, which is complex according to the available data. Hoffmans Advice can help you take a step forward in assessing dikes and hydraulic structures, especially for the piping process.

Differences between the Shields–Darcy model and Sellmeijer model
Stability and piping berm built during high water in 1995 (Rijkswaterstaat)

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