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ABSTRACT
This article discusses the piping erosion time, where pipes start to erode in aquifers underneath dikes and dams until they reach their critical values. The magnitude of the piping erosion time significantly determines the risk of failure of water defences. The time-scale equation is based on a sediment mass balance equation and appropriate bedload transport predictors, assuming the erosion process to be continuous. We argue that the flow is laminar for pipes in sandy aquifers and turbulent for pipes in gravel aquifers. We then account for aquifer composition in examining pipe erosion by discussing different bedload transport predictors for each flow regime. To estimate the turbulence intensity, we have used and modified the Einstein bedload transport theory. The time-scale relation includes the effects of meander bends and has been tested for some experiments on a small scale and on a large scale.
Acknowledgements
The author thanks, in particular, Prof. Dr. Wim Uijttewaal (TU Delft Chair of Experimental Hydraulics), emeritus Prof. Dr. Leo van Rijn, Dr. Erik Mosselman (river engineering specialist of Deltares), Henk Verheij (hydraulic specialist) and the anonymous reviewers for their advice, discussion and feedback.
Disclosure statement
No potential conflict of interest was reported by the author.
Notation
| c | = | piping erosion celerity (m s−1) |
| Ce | = | Fell coefficient or soil erosion parameter (s m−1) |
| d = d50 | = | mean particle size (m) |
| D | = | thickness of the sand layer (m) |
| g | = | acceleration of gravity (m s−2) |
| kd | = | (Ce/ρdry) Temple and Hanson coefficient of erosion or soil erosion parameter (s m2 kg−1) |
| kN | = | Nikuradse roughness (m) |
| k0 | = | depth-averaged turbulent kinetic energy (m2 s−2) |
| K | = | hydraulic conductivity (m s−1) |
| ℓ | = | pipe length (m) |
| ℓp | = | mean pipe height or pipe height halfway along the pipes (m) |
| L | = | seepage length (m) |
| n | = | porosity (−) |
| p | = | PDF = probability density function (−) |
| P | = | probability or CDF = cumulative density function (−) |
| qb | = | bedload transport per unit width (m2 s−1) |
| r0 | = | mean relative turbulence intensity (−) |
| R | = | hydraulic radius (m) |
| Re | = | (R Up/ν) Reynolds number (−) |
| Re,part | = | (d Up/ν) particle Reynolds number (−) |
| Re* (= d u*/ν) | = | particle Reynolds number of Nikuradse (−) |
| Sdike | = | hydraulic dike gradient (−) |
| Spipe | = | hydraulic pipe gradient (−) |
| t | = | time (s) |
| tp | = | piping erosion time (s) |
| T ′ | = | (µτ /τc) instantaneous transport parameter (−) |
| u* | = | bed/wall shear velocity (m s−1) |
| Up | = | mean flow velocity or pipe flow velocity halfway along the pipes (m s−1) |
| vp | = | propagation rate (m s−1) |
| W | = | width of the flume (m) |
| x | = | longitudinal coordinate (m) |
| z | = | vertical coordinate (m) |
| α | = | coefficient (−) |
| β | = | coefficient (−) |
| ϕ | = | dimensionless bedload transport (−) |
| µ | = | roughness factor (−) |
| ρ | = | density of water (kg m−3) |
| ρdry | = | dry density of sediment (kg m−3) |
| ρs | = | density of sediment (kg m−3) |
| στ | = | standard deviation of τ (N m−2) |
| τ | = | instantaneous bed/wall shear stress (N m−2) |
| τ0 | = | mean bed/wall shear stress (N m−2) |
| ν | = | kinematic viscosity (m2 s−1) |
| Ψ | = | dimensionless bed shear stress or Shields parameter (−) |
Subscripts
| c | = | critical |
| ℓam | = | laminar |
| max | = | maximum |
| tur | = | turbulent |