1 INTRODUCTION
The internal erosion of a sandy aquifer below a river dike, a process called piping, is a dominant failure mechanism in the Netherlands. The occurrence of piping can be estimated using computation models. The models that are used most in the Netherlands are the empirical relation of Bligh (Bligh, 1910) and the semi-analytic model of Sellmeijer (Sellmeijer, 1988). The model of Sellmeijer is currently the most ad-vanced method, as it predicts the critical hydraulic head across the dike, taking into account both geo-metrical parameters and sand characteristics.
1.1 Sellmeijers model (Sellmeijer, 1988)
The model of Sellmeijer is based on the equilibrium of forces of sand grains, flow in the channel (pipe flow) and the flow through the aquifer (Darcy). The model can be described by three factors: resis-tance, scale and geometry (Equation 1)
tan 70 3 tan( ) ( ) 3 geometry resis ce scale p c w F F F H d F G L L (1) In which:
Hc = critical head fall across the levee
’p = volumetric weight of sand under water [kN/m3] w = volumetric weight of water [kN/m3]
= bedding angle [ ]
= coefficient of White [-]
= intrinsic permeability of the sand layer [m2] d70 = grain size at 70% cumulative weight fraction [m] F(G) = factor related to the geometry of the sand body [-] L = seepage length [m]
In this equation the factor related to the geometry is dependent on the ratio between thickness and seep-age length, which must be determined numerically, by applying an erosion computing model (MSeep).
The d70 is considered the representative grain size
for the piping process, which is linearly incorporated in the rule. Other included sand characteristics are the permeability and bedding angle. The bedding an-gle, which defines the resistance of sand against roll-ing, however, has not been defined for different sand types.
In order to validate this computation model, and to calibrate it for application purposes, large-scale tests have been conducted (Silvis, 1991), however very limited in number, due to the high costs in-volved. Only one sand type has been used in these tests, therefore the explicit validation of the critical head fall as function of the relevant sand characteris-tics in the computation model could not be achieved. After the validation, the theoretical bedding angle is adjusted to the results of the tests.
Influence of sand characteristics and scale on the piping process –
experiments and multivariate analysis
V.M. van Beek
Deltares/Delft University of Technology, Delft, The Netherlands
J.G. Knoeff, J. Rietdijk, J.B. Sellmeijer, J. Lopez De La Cruz
Deltares, Delft, The Netherlands
ABSTRACT: Recent studies for levee safety in the Netherlands gave rise to a more detailed investigation of the phenomenon of piping (retrograde internal erosion in aquifers below levees). As part of an extensive re-search program, small-scale experiments (L=0.50m) have been performed to gain insight into the characteris-tics of sands that influence the process of piping, possibly leading to levee failure. In about fifty experiments, different sands with varying properties (grain size distribution, grain angularity) have been tested at various relative densities. In these experiments, the hydraulic head over a sand sample has been increased until piping occurred. To assess the influence of scale on the observed processes, medium-scale experiments have been per-formed with larger dimensions (L=2m) on two types of sands at various relative densities. Based on the results of the small- and medium-scale experiments an adjustment for the prediction rule has been suggested, using multivariate analysis.
1.2 Research objectives
In this research the influence of sand characteristics, like relative density, grain size, grain size distribution, grain angularity and permeability, have been investi-gated in small-scale experiments. As most of the sand characteristics cannot be varied exclusively, nine sand types with varying properties have been tested at a relative density of around 70%. Two sand types have been tested at different relative densities.
In the model of Sellmeijer a length scale effect is included. Due to this scale effect the critical gradient decreases with increasing seepage length. To validate the scale effect and to confirm the results of the small-scale experiments at different scales, medium-scale experiments have been performed. These ex-periments are similar to the small-scale exex-periments, but are a factor 4 larger in each dimension.
To distinguish the influence of the involved pa-rameters for interpretation a multivariate analysis has been performed. Such an analysis provides the expo-nential influence of the parameters for the particular tests carried out. Of course, full scale tests are re-quired to confirm the results of the experiments and multivariate analysis.
This research is part of a larger research program called Strength and Loading of Flood Defence Struc-tures (SBW), in which improvement of prediction models of different failure mechanisms for levees is pursued. SBW Piping specifically focuses on the im-provement and applicability of prediction for piping, to improve testing methods for the 5-yearly safety assessment of Dutch levees.
2 EXPERIMENTAL SET-UPS
2.1 Small-scale experiments
To determine the influence of the sand characteristics on the piping process, a series of small–scale experi-ments has been performed. The set-up consisted of a pvc box with dimensions of 0.5x0.3x0.1m. In vertical position, the box can be filled with sand and the rela-tive density can be controlled. After placing the box in the horizontal position, the sand is retained by two filters. A constant head can be applied to the sand, with a range of 0-1m. The transparent Perspex cover allows observation of the formation of piping chan-nels.
Figure 1. Schematization of the set-up of the small-scale ex-periments
A set of experiments was conducted to investigate the influence of sand characteristics. In these experi-ments nine types of sand were tested, which repre-sent a variety of Dutch sands. The sands are of dif-ferent origin and differ in properties of grain size, grain size distribution, permeability and angularity. All tests are performed at a relative density of around 70% and are performed twice, for reproducibility.
The used sand types and properties are listed in table 1. The names of the sand types refer to their origin places and are not categorized into interna-tionally adopted sand characteristics. The grain size distribution is defined by the uniformity coefficient (U=d60/d10). The angularity is defined by the
KAS-value (based on Powers (1953)). This KAS-value is visu-ally obtained, based on Figure 2. Most of the fine fraction has been removed from the sand before preparation.
Figure 2. KAS- indication of angularity
Table 1. Sand characteristics (small-scale experiments)
Name d70 U (d60/d10) KAS µm - - Nunspeet Dekzand 192 2.6 54 Enschede 307 2.1 51 Hoherstall Waalre 400 1.6 46 Hoherstall Sterksel 232 2.2 35 Itterbeck Scheemda 175 1.3 38 Itterbeck Enschede 431 1.6 69 Itterbeck Sandr 195 1.5 52 Itterbeck Dekzand 202 2.2 45 Baskarp 154 1.6 50
Nunspeet and Baskarp sands have been tested more extensively than the other types of sand: for these types of sands the relative density has been varied to from 35% to 95% (in an extra series of duplicate tests) to investigate the influence of relative density on the piping process.
The sand has been prepared using the ‘wet method’, in which dry sand is scattered into water with the box in vertical direction. Densification takes place either by giving a pulse to the filled sample box (lifting and dropping) or by continuous tamping dur-ing the scatterdur-ing of sand. The former method is suit-able for creating relative densities up to 70%. The latter method is suitable for creating very dense sam-ples, from 80 to 100%. These methods ensure a ho-mogeneous and well-saturated sample.
The experiments have been performed by increas-ing the hydraulic head with 1 cm per 5 minutes, until channel formation takes place. In case of channel formation, the increase of hydraulic head was delayed until the pipe was stable for at least two minutes. The result which is obtained from the experiment is the
hydraulic head at breakthrough (denoted as Hc,exp),
which is the head at which the pipe has grown from the downstream side towards the upstream side, such that a connection is present between the upstream and downstream head. Theoretically there should be a very small difference between the ‘head at break-through’ and the ‘critical head’, which is obtained from calculations with the model of Sellmeijer.
2.2 Medium-scale experiments
Medium-scale experiments have been performed to investigate scale effects and to verify the results from the small-scale experiments.
The set-up of the medium-scale experiments is very similar to the set-up of the small-scale experi-ments. The dimensions of the sand container are four times bigger in length and depth (inner dimensions 2*0.88*0.40 m).
Figure 3. Set-up (medium-scale experiments)
The sand is prepared in a similar way as in the small-scale experiments, by placing the container in vertical direction. Due to the large dimensions of the con-tainer, the sand is filled and removed in a more effi-cient way, which is described in (Rietdijk et al., 2010). After sand preparation the set-up can be put in horizontal position for the execution of the test.
The sand types used are Baskarp sand and Itter-beck sand (fraction of 125-250 m). The properties of both sands are listed in table 2. Based on the re-sults of the small-scale experiments, KAS and uni-formity appear to be of lesser importance than the other sand characteristics, and are therefore not var-ied. Both sand types have been tested at a relative density of 50% and 75%. All experiments have been duplicated.
Table 2. Sand characteristics (medium-scale experiments d70 U (d60/d10) KAS
µm - -
Itterbeck (125-250) 210 1.7 52
Baskarp 154 1.6 50
During the experiments a gradient is gradually ap-plied to the sand. Every five minutes, the hydraulic head is increased with 2 cm during the first 10 cm of head fall and 1 cm until sand transport is started. In case of channel formation, the increase of hydraulic head was delayed until the pipe was stable for at least two minutes. The head at breakthrough, to be com-pared with the critical head, is defined in the same way as in the small-scale experiments.
3 RESULTS AND DISCUSSION
3.1 Small-scale experiments
In the small-scale experiments nine types of sands have been tested at a relative density of 70%. In these experiments many different patterns of erosion have been found. Most of the channels started at the downstream side of the sand sample, some started in the middle. In the beginning of the process, pipes may be visible at various locations. After a while, one pipe will become dominant. Sometimes pipes split into different branches. It is unknown which sand characteristics determine the erosion pattern. In fig-ure 4 one of the processes is shown.
Figure 4. Example of erosion process in small-scale experi-ment
The differences between the theoretical critical head and experimental head at breakthrough can be seen in figure 5. A large scatter is present in the data.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 Hc_exp/L [-] H c /L [ -]
Figure 5. Theoretical Hc/L vs. experimental Hc/L (small-scale experiments)
The experiments showing the largest differences be-tween theoretical critical head and experimental head at breakthrough are the ones that are performed on sands with larger grain size. Therefore, at first glance, grain size seems to be the dominant parame-ter that causes the largest deviations from the model. Whereas a positive trend is expected based on the model of Bligh and Sellmeijer, a negative trend is found in the experiments (Figure 6). The negative trend is not yet fully understood.
0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,0 100,0 200,0 300,0 400,0 500,0 d70 [ m] H c /L [ -] Calculation Experiment Linear (Experiment) Linear (Calculation)
Figure 6. Theoretical and experimental Hc/L vs. D70
For two types of sand (Baskarp sand and Dekzand), the relative density has been varied. In the experi-ments at low relative density (35%) it was found that a different process occurred. It is unknown to which extend this process may occur in practice and further investigation appears necessary. These experiments have therefore been excluded from further analysis.
At medium and high relative density experiments (>50%) retrograde erosion occurred. As permeability decreases with increasing density, theoretically an in-crease of hydraulic head was expected with increas-ing relative density. In figure 7 the relative density has been plotted against the theoretical critical head and the experimentally obtained head at break-through for the experiments on Baskarp sand in which retrograde erosion took place.
0 0,1 0,2 0,3 0,4 0,5 0,6 40 50 60 70 80 90 100 RD [%] H c /L [ -] Experiment Calculation
Figure 7. Theoretical and experimental Hc/L vs. relative den-sity
For these experiments a positive correlation appears to be present between relative density and hydraulic head. The results comes up tot the expectations.
3.2 Medium-scale experiments
Eight medium-scale experiments have been per-formed on Baskarp and Itterbeck sand. One of the experiments has been removed from the data set due to large deviation of permeability, forward erosion and break-down of the pump.
In the other experiments the erosion started in the middle of the sand sample or at the downstream side (retrograde) erosion. Generally more than one chan-nel formed, creating a pattern of chanchan-nels next to each other.
The results of the seven experiments are shown in figure 8, in which the predicted experimental gradient at breakthrough is plotted against the calculated critical gradient, in which large deviations can be ob-served. The largest deviations are found in the ex-periments performed on Itterbeck sand.
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,0 0,1 0,1 0,2 0,2 0,3 0,3 Hc_exp/L [-] H c /L [ -]
Figure 8. Theoretical Hc/L vs. experimental Hc/L (medium-scale experiments)
One purpose of the medium-scale experiments was to verify the results of the small-scale experiments. In Figure 9, in which the critical gradient is plotted against the grain size (d70), it can be observed that,
the number of data points is too limited to be conclu-sive on possible relationships between Hc and d70.
0 0,05 0,1 0,15 0,2 0,25 0,3 0,0 100,0 200,0 300,0 400,0 500,0 d70 [ m] H c /L [ -]
Calc ulat ion Ex periment Li near ( Ex periment ) Linear ( Calculat ion)
Due to the few data points and limited accuracy of the determination of relative density in the medium-scale set up, the influence of this parameter could not be verified. However, the negative trend between d70
and gradient as is found in the small-scale experi-ments is not contradicted either.
The second purpose of the experiments was to in-vestigate the length scale effect. In order to deter-mine the influence of length scale, the medium-scale experiments performed on Baskarp sand have been compared with small-scale experiments. In figure 10 the results of the Baskarp sand experiments (in which relative density is around 60%) and their predictions have been plotted against the seepage length. It ap-pears that the critical gradient is correctly predicted for both of the seepage lengths.
0,000 0,100 0,200 0,300 0,400 0,500 0 0,5 1 1,5 Length [m] H c /L Experiment Calculation
Figure 10. Influence of length scale
In 1991 the influence of length scale has been inves-tigated at larger scale already (Silvis, 1991). Based on these experiments the model of Sellmeijer had been calibrated. Considering the fact that for the small-scale, medium-scale and these large scale ex-periments the model predicts the critical gradient well, it is concluded that the scale effects are cor-rectly modeled.
3.3 Multivariate analysis
As it was impossible to vary all parameters exclu-sively, in the small-scale experiments nine different sands have been chosen with varying properties. To interpret the influence of the different parameters, a multivariate analysis has been performed using re-gression analysis.
For the multivariate analysis a selection has been made of the data set. As the model of Sellmeijer is based on retrograde erosion, only the experiments that have shown this process are included. This leaves out especially some of the experiments per-formed at a low relative density. Some of the ex-periments were marked as outliers and have been re-moved from the data set as well. Finally a number of 38 experiments is left for the analysis.
For the analysis it has been assumed that the pa-rameters of relative density (RD), coefficient of uni-formity (U), angularity (defined by KAS),
permeabil-ity, grain size (represented by d70) may be of
influence on the piping process. The empirical for-mula as shown in Equation 2 is therefore used as in-put for the analysis:
70 m m m m m 70m c H RD U KAS d H L RD U KAS d L (2)
In this equation the critical gradient is related to the relevant parameters. The parameters have been di-vided by the mean value of the data set (indicated by the subscript ‘m’) to equalize the values of the differ-ent compondiffer-ents. In this way, the compondiffer-ents have become dimensionless. The influence of the five di-mensionless components can be determined by calcu-lating the values of , , , and by means of re-gression analysis. Based on the analysis of the data set of the small-scale experiments the following em-pirical relation can be deduced:
0.39 0.35 0.13 0.02 0.35 70 m m m m m 70m c H RD U KAS d H L RD U KAS d L (3)
The multivariate regression results show that the relative density, permeability and grain size (d70) have
a strong influence on the critical gradient. The uni-formity coefficient and the angularity of the grains (KAS) seem to have a weak influence in the results. However, since some of the parameters have a strong connection among themselves it is important to re-mark that the analysis will record this connection, giving more weight to one of the parameters and less to the other. Therefore analyses have been made in which one of the parameters has been excluded. The results suggest that the variables that have the high-est impact in the critical head are the relative density, the permeability and the d70. However, a further
revi-sion of the relationship between the experimental values for the uniformity coefficient and the d70 is
recommended.
In Figures 11 and 12 the curve fit of the multivari-ate analysis and the model of Sellmeijer are shown. The error, obtained with the regression analysis, is decreased from 20.49 to 0.3. -1.5 -1 -0.5 0 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 ln(Hc/L) [-] m u lt iv a ri a te r e g re s s io n error=0.3
0 5 10 15 20 25 30 8 10 12 14 16 18 20 22 24 Hc [cm] H c S e llm e ije r [c m ] residual=20.49
Figure 12. Curve fit for the predictions using the model of Sellmeijer
3.4 Comparison with and extension of current
model
The current model of Sellmeijer already includes some of the relevant parameters, such as permeability and grain size. Other parameters, like angularity, uni-formity and relative density are not yet included.
The influence of permeability as obtained from the multivariate analysis is very similar to the way it is incorporated in the current model. The influence of grain size is linearly present in the current model. The multivariate analysis indicates that the influence in the small-scale experiments is d70
0.394
. This discrep-ancy cannot yet be theoretically explained.
An adjustment of the model may be realized by extending the model with parameters of influence like relative density and uniformity coefficient. The ad-justment of the influence of grain size may be real-ized by an extension to the scaling factor.
The limited number of medium-scale tests is insuf-ficient to be able to perform a good multivariate analysis. The data can be used to validate an adjusted model of Sellmeijer. It is noted that an adjustment of the model will not be valid for practice unless full-scale validation has taken place.
4 SUMMARY AND CONCLUSIONS
A large number of small- and medium-scale experi-ments have been performed to assess the influence of sand characteristics, like grain size, uniformity coeffi-cient, angularity, permeability and relative density and length scale effects on the piping process.
From the small-scale experiments it was found that retrograde erosion, as is posed by the model of Sellmeijer, occurs at relative densities larger than 50%. At lower densities a different process may oc-cur, for which further research is required.
A multivariate analysis has been performed on the results of the small-scale experiments showing retro-gade erosion to interpret the influences of the differ-ent parameters. It was found that the influence of
permeability as obtained from the analysis is similar to the way it is incorporated in the theoretical model. The grain size appears to be of less importance in the experiments than in the theoretical model, causing an increasing discrepancy with increasing grain size. The influence of angularity on the pipingproces appears to be limited. Uniformity coefficient of sand contrib-utes slightly. Relative density appears to be an impor-tant parameter to include in the model.
Medium-scale experiments have been performed to assess scale effects and to verify the results of the small-scale experiments. Comparison of the experi-ments with theory shows that scale effects are well predicted by the model of Sellmeijer. The decreased influence of grain size, as found in the small-scale experiments, is not contradicted by the medium-scale experiments. The influence of relative density could not be verified, as the medium-scale set up did not al-low for accurate relative density determination.
It is suggested to adjust the model of Sellmeijer, taking into account the results of the multivariate analysis. This can be done by extending the formula with relative density and by diminishing the influence of d70. However, before the model can be accepted
for practice, full-scale validation is necessary. It is also recommended to find a theoretical explanation for the deviating influence of grain size. Next to this, the multivariate results are based on a data set con-sisting of experiments on prevailing Dutch sands. Outside this range, results may deviate.
5 ACKNOWLEDGEMENTS
The research was funded by Rijkswaterstaat – Centre for Water Management – on behalf of the Ministry for Transport, Public Works and Water Manage-ment. This funding was realised in the framework of the research program “Strength and Loading on Flood Defence Structures”.
REFERENCES
Bligh, W.G. (1910). “Dams Barrages and weirs on porous foundations” Engineering News 64 (26).
Powers, M. C. (1953). “A new roundness scale for sedimen-tary particles” Journal of Sedimensedimen-tary Petrology, v. 23, p. 117-119
Rietdijk, J., Schaminée, P.E.L., Schenkeveld, F.M. (2010), “The drizzle method for sand sample preparation”, 7th In-ternational Conference on Physical Modelling in Geotech-nics, 2010 Zurich.
Sellmeijer, J. B. (1988). "On the mechanism of piping under impervious structures," Delft University of Technology, Delft.
Silvis, F. (1991). “Verificatie Piping Model; Proeven in de Deltagoot. Evaluatierapport” Rapport Grondmechanica Delft, CO317710/7.
