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In unstable soils, a special erosion process termed suffusion can occur under the effect of relatively low hydraulic gradient. The critical hydraulic gradient of an unstable soil is smaller than in stable soils, which is described by a reduction factor α. According to a theory of Skempton and Brogan (1994) [1], this reduction factor is related to the stress conditions in the soil. In an unstable soil, the average stresses acting in the fine portion are believed to be smaller than the average stresses in the coarse portion. It is assumed that the stress ratio and the reduction factor for the hydraulic gradient are almost equal. In order to prove this theory, laboratory tests and discrete element modelings are carried out. Models of stable and unstable soils are established, and the stresses inside the sample are analysed. It is found that indeed in unstable soils the coarse grains are subject to larger stresses. The stress ratios in stable soils are almost unity, whereas in unstable soils smaller stress ratios, which are dependent on the soil composition and on the relative density of the soil, are obtained. A comparison between the results of erosion tests and numerical modeling shows that the stress ratios and the reduction factors are strongly related, as assumed by Skempton and Brogan (1994) [1].

For the subsoil below pavements, dams or dikes, the stability against erosion with regard to seepage forces induced by under-seepage flow has to be proved. In that respect, it is of particular importance to identify and assess unstable soils, because here at relatively small hydraulic gradients, a special erosion process termed “suffusion” can occur. In general, gap-graded or well-graded soils are sensitive to suffusion. As result of the process, the fine fraction of the soil is washed out through the pores between the coarse-grained soil fraction. This increases the soil permeability and can lead to large civil works settlements or to the failure.

To assess whether suffusion is possible, in general the composition of the soil and the geometry of the pore channels have to be considered. Suffusion is only possible if the grains of the fine soil can pass through the pores of the coarse soil matrix. Since the pore channel geometry cannot be exactly measured, the assessment is based on the curve of the grain size distribution, which is related to the pore channel geometry. According to the Kenney and Lau (1986) criterion [_{min} derived from the grain size distribution shall be less than 1.0. Here, H is the masse percent of soil between the grain diameter d and 4d. F is the mass percent of grain with diameter smaller than d. The authors of the paper in hand proposed to derive a parameter (d_{c,15}/d_{f,85})_{mod} from the grain size distribution and to use (d_{c,15}/d_{f,85})_{mod} < 4 as a stability criterion, Ahlinhan (2011) [

If the “geometric” criterion yields the result that suffusion is possible, i.e. the soil is potentially unstable the minimum hydraulic gradient necessary to cause erosion and to transport the fine soil grains has to be assessed by

a “hydraulic” criterion. In stable soils, the critical hydraulic gradient for vertical upwards seepage flow

with:

In unstable soils, the respective hydraulic gradient is generally smaller. This fact can be described by a stress reduction factor α. Skemptonand Brogan (1994) [

and

with:

In this paper, the results of experimental tests regarding critical hydraulic gradients for upwards seepage in different soils with varying relative densities are presented. Also discrete element models are performed to determine the stress ratios in these soils. By comparison of the results, the validity of the Skempton and Brogan [

Five different non-cohesive soils were tested in a specially developed test device under vertical upward seepage flow. The hydraulic gradient was increased slowly and gradually in order to identify the critical gradient at which erosion begins. The initial relative density of the soils was varied in the laboratory tests. The used relative

density is defined as_{max} is the maximum porosity, n_{min} is the minimum porosity and n is the actual porosity of the soil.

The grain size distributions of the five soils are shown in _{min} = 4.4 and 5.93). The soils E1, E2 and E3 are gap-graded soils, which were produced artificially. E2 and E3 are clearly unstable soils, whereas E1 has an (H/F)_{min} value of 1.1 and so lies on the border between the stable and unstable region with regard to the Kenney and Lau (1986) [_{c,15}/d_{f,85})_{mod} < 4 also leads to a close decision regarding internal stability.

A photographic view and a schematic drawing of the test device for vertical upward flow are shown in

Details regarding sample preparation, execution and evaluation of the tests can be found in Ahlinhan and Achmus (2011) [

For the stable soils A1 and A2 the obtained critical gradients agree quite well with the theoretical values according to Terzaghi [

For the clearly unstable soils E2 and E3 very small critical gradients between 0.18 and 0.23 were measured. There is only a small dependence on the relative density of the sample. On the contrary, for soil E1, which is on

Property | Soils | ||||
---|---|---|---|---|---|

A1 | A2 | E1 | E2 | E3 | |

Density of grains γ_{s} [t/m^{3}] | 2.65 | 2.65 | 2.65 | 2.65 | 2.65 |

Minimum porosity n_{min} | 0.40 | 0.32 | 0.34 | 0.27 | 0.31 |

Maximum porosity n_{max} | 0.52 | 0.43 | 0.42 | 0.40 | 0.42 |

Coefficient of uniformity C_{u} | 2.10 | 3.00 | 7.00 | 13.90 | 23.40 |

Index of curvature C_{c} | 1.00 | 1.00 | 3.30 | 6.70 | 13.80 |

(H/F)_{min} | 5.93 | 4.44 | 1.10 | 0.20 | 0.03 |

(d_{c,15}/d_{f,85})_{mod} | 1.31 | 1.50 | 3.30 | 7.20 | 14.40 |

the border between stable and unstable, a stronger dependence of the critical hydraulic gradient on the relative densities was found. For a very dense state the critical hydraulic gradient is about double the value determined for a medium dense state. However, the value for very dense state is also significantly smaller than the theoretical critical hydraulic gradient as per Terzaghi [

_{min} and once on (d_{c,15}/d_{f,85})_{mod}. The values obtained by Skempton and Brogan (1994) [_{min} suggested by Skempton and Brogan (1994) [_{c,15}/d_{f,85})_{mod}, but the scatter here is slightly larger. The trend lines in

The determination of effective stresses acting in the fine and coarse portions of the soils A1, E1, E2 and E3 given in _{n}, tangential stiffness k_{s}) and the Coulomb friction law (friction coefficient μ_{s}). The maximum shear force F_{smax} is related to the normal force F_{n} by F_{smax} = μ_{s}∙F_{n}. (

For the numerical simulation of the triaxial test the particles were generated using the “fill and expand” method (Itasca 2003 [

During the calibration process first the model parameters k_{n} (normal stiffness) and k_{s} (tangential stiffness) were varied in order to match the stress-strain curve of real soil for strains lower than 1%, i.e. the deformation modulus, while the other parameters were kept constant. Then the inter-particle friction µ_{s} was varied to adjust the peak failure stress. The mobilized interne friction angle according to Equation (4) and the volumetric strain development of soil A1 are shown in

As can be seen in _{mob} and the simulated triaxial tests by using k_{n} = k_{s} = 10^{5} N/m, µ_{s} = 2 for soil A1. The volumetric strain results show that the discrete element assembly exhibits more initial compression and more subsequent dilation than the real soil. This result corresponds to those observed in similar numerical simulations by tom Woerden et al. (2004) [

In a similar way, the parameters for the soil E2 have been calibrated on triaxial tests carried out with this soil. Here, the parameters k_{n} = k_{s} = 10^{5} N/m, µ_{s} = 3.2 have been found to give optimum agreement of experimental and numerical results. The same set of parameters has been used for the other unstable soils E1 and E3.

In order to keep the calculation effort in reasonable limits, the assessment of the reduction factor α was carried out with a two-dimensional model, i.e. using the PFC^{2D}software. A numerical model of 2.50 cm × 3.75 cm was developed in PFC^{2D} (_{0}-state” (σ_{h} = k_{0}∙σ_{v} with k_{0} = 1 − sinφ’), a small reduction of particle sizes was carried out and the redistribution of particles was again calculated until equilibrium was found. In all cases, the required particle size reduction was less than 0.1%.

To achieve different relative densities of the resultant sample, a method described in detail in tom Woerden et al. (2004) [_{s} = 0 was used during sample generation, and to achieve a “loosest” (D = 0.0) sample, the value was set to μ_{s} = 15. Intermediate μ_{s}-values were then used to achieve relative densities of 0.25, 0.50, 0.75 and 0.95.

The final step was the calculation of the reduction factor α. To do this, the intersected particles at a cross section through a defined depth of the sample were identified. The effective stresses in each particle were determined with a special calculation routine using the “Fish” macro language in PFC Itasca 2003 [_{min} according to the Kenney and Lau (1986) [

It was found that the α-values are dependent on the considered depth of the cross section. This is shown by the results for the soils A1 and E2 given in

Regarding the random generation of particles, only minor effects on the determined α-value were found.

Soil A1 | Soil A2 | |||||
---|---|---|---|---|---|---|

Depth [cm)] | 1.00 | 1.80 | 2.80 | 1.00 | 1.80 | 2.80 |

Reduction factor α [-] | 0.80 | 0.96 | 0.94 | 0.11 | 0.10 | 0.14 |

Soil A1 | Soil A2 | |||||
---|---|---|---|---|---|---|

Number of random generation | N˚1 | N˚2 | N˚3 | N˚1 | N˚2 | N˚3 |

Reduction factor α [-] | 0.96 | 0.98 | 0.97 | 0.10 | 0.09 | 0.09 |

In

As mentioned in Section 2, an instability index (d_{c,15}/d_{f,85})_{mod} can be used to formulate an instability criterion. In

From the experiments, reduction factors were determined as follows:

These experimental results, which are related to hydraulic gradients, are compared to the numerical simulation results related to stress ratios.

Experimental tests presented in the paper in hand show that the critical hydraulic gradient for upward seepage flow is dependent on the instability index of the soil and to a minor extent also on its relative density. According to a hypothesis formulated by Skempton and Brogan (1994) [

This hypothesis is checked by means of numerical simulations with the discrete element method and laboratory tests. Different soils are considered, and the model parameters are calibrated by comparison with the results of triaxial tests. It is found that the non-linear behavior of non-cohesion soil can be simulated fairly well by means of discrete element method. With the numerical model, stress ratios describing the non-homogeneity of stress conditions can be derived.

Good agreement is obtained between reduction factors regarding hydraulic gradients stemming from experimental tests and the stress reduction factors obtained in the numerical simulations. These results indicate that there is indeed a strong relation of these factors and thus strongly support the hypothesis of Skempton and Brogan (1994) [

This research was partially supported by the Germany Ministry of Education and Research through IPSWaT (International Postgraduate Studies in Water Technologies). This support is gratefully acknowledged.

Marx Ferdinand Ahlinhan,Emmanuel Kokou Wouya,Yvette Kiki Tankpinou,Marius Bocco Koube,Codjo Edmond Adjovi, (2016) Experimental and Numerical Investigation of Stress Condition in Unstable Soil. Open Journal of Civil Engineering,06,370-380. doi: 10.4236/ojce.2016.63031