Evaluation of Combined Base Soil Erodibility and Granular Filter Efficiency

Abstract

Filters managed in zoned dams are designed according to criteria based on the grain size distribution of both filter and eroded soil. However, the constriction size distribution of the filter is the key parameter which governs the filter retention process of flowing eroded particles. To assess the filter efficiency regarding eroded particles, several filters and base soils are tested in a vertical cell with a configuration coupling erosion and filtration processes. For setting the boundary condition of eroded particles at the filter inlet, hole erosion test (HET) was performed on the base soil. The investigation of the evolution of filter behavior shows that the void ratio and the grain shape are of a great influence on filter efficiency. A new approach of filter clogging was proposed by evaluating a damage index which is affected by various parameters such as the ratio D₁₅/d₈₅ and the size of eroded particles. An approach linking the geometrical parameters (damage index) to the hydraulic conductivity leads to an estimation of the filter performance which provides a more quantifiable and realistic criterion. The results indicate that even existing criteria were not met; the tested filters remain efficient as regards to experimental data. An analytical approach based on constrictions size distribution was used and pore reduction was matched with experimental results.

Appendix: Analytical Model of the Constriction Size Distribution

The most common definition of the pore size is the diameter of the inscribed sphere between tangential particles. Comparisons with the void volumes are made possible by associating a constriction diameter with each pore. The analytical approach aims to compute the CSD from essential information about the PSd of the granular material. It consists of applying a probabilistic schema to an assumed geometrical packing structure within the filter. A further basic concept based on a probabilistic approach is shown in Fig. 9 for two cases, i.e., the loosest and densest filters. The fine particles sequentially flow from one pore to another, passing through the constrictions. Actually, the occurrence of a given constriction size is related to the possibility of particle contact to create such a constriction. This design criterion was based on the probability of the grains’ arrangement in the filter matrix in order to form the largest voids. The size of these voids is dependent on the size and packing geometry of the filter particles.

Fig. 9

Constructions Size Distribution of a material: (a) loose case, (b) dense case

Silveira et al. (1975) assumed that for the densest geometric configuration, the constriction size D_c3 is made up of three tangent spheres of diameters D_i, D_j, D_k. The size of Dc₃ can be deduced as follows:

$$\left( {\frac{2}{{{\text{D}}_{{\text{i}}} }}} \right)^{2} + \left( {\frac{2}{{{\text{D}}_{{\text{j}}} }}} \right)^{2} + \left( {\frac{2}{{{\text{D}}_{{\text{k}}} }}} \right)^{2} + \left( {\frac{2}{{{\text{D}}_{{{\text{c}}3}} }}} \right)^{2} = \frac{1}{2}\left( {\frac{2}{{{\text{D}}_{{\text{i}}} }} + \frac{2}{{{\text{D}}_{{\text{j}}} }} + \frac{2}{{{\text{D}}_{{\text{k}}} }} + \frac{2}{{{\text{D}}_{{{\text{c}}3}} }}} \right)$$

(8)

The probability of the occurrence P_c3 by the surface of the constriction size D_c3 is provided by Eq. 9.

$${\text{P}}_{{{\text{c}}3}} = \frac{3!}{{{\text{r}}_{{\text{i}}} !{\text{r}}_{{\text{j}}} !{\text{r}}_{{\text{k}}} !}}{\text{p}}_{{\text{i}}}^{{{\text{r}}_{{\text{i}}} }} {\text{p}}_{{\text{j}}}^{{{\text{r}}_{{\text{j}}} }} {\text{p}}_{{\text{k}}}^{{{\text{r}}_{{\text{k}}} }}$$

(9)

where r_i, r_j and r_k are the numbers of times that the diameters D_i, D_j and D_k appear in the three particle groups, respectively; r_i, r_j, r_k = 0, 1, 2, 3 and r_i + r_j + r_k = 3; p_i, p_j and p_k are the percentage (probability of occurrence by area) of D_i, D_j and D_k, respectively.

It was assumed that in the loosest state the area S_V formed by four tangent spheres with respective diameters D_i, D_j, D_k and D_m and respective probabilities of occurrence by area p_i, p_j,p_k and p_m (Fig. 1a), the constriction size D_c4 can be expressed by Eq. 10.

$${\text{D}}_{{{\text{c}}4}} = \sqrt {\frac{{4{\text{S}}_{{{\text{VMAX}}}} }}{{\uppi }}}$$

(10)

where S_Vmax is the maximum area formed among the four tangent particles. The probability of the occurrence by area P_c4 of the constriction size D_c4 is computed from Eq. 11 as follows:

$${\text{P}}_{{{\text{c}}4}} = \frac{4!}{{{\text{r}}_{{\text{i}}} !{\text{r}}_{{\text{j}}} !{\text{r}}_{{\text{k}}} !{\text{r}}_{{\text{m}}} !}}{\text{p}}_{{\text{i}}}^{{{\text{r}}_{{\text{i}}} }} {\text{p}}_{{\text{j}}}^{{{\text{r}}_{{\text{j}}} }} {\text{p}}_{{\text{k}}}^{{{\text{r}}_{{\text{k}}} }} {\text{p}}_{{\text{m}}}^{{{\text{r}}_{{\text{m}}} }}$$

(11)

where r_i, r_j, r_k and r_m are the numbers of times that the diameters D_i, D_j, D_k and D_m appear in the four particles groups, respectively; r_i, r_j, r_k, r_m = 0, 1, 2, 3, 4 and r_i + r_j + r_k + r_m = 4; p_i, p_j, p_k and p_m are the percentage (probability of occurrence by area) of D_i, D_j, Dk and D_m respectively.

The constriction area S_V can be computed (Fig. 9) from Eq. 12 below:

$${\text{S}}_{{\text{V}}} = \frac{{{\text{ad}}\sin \alpha }}{2} + \frac{{{\text{bc}}\sin \gamma }}{2} - \frac{1}{8}\left( {{\text{D}}_{{\text{i}}}^{{\text{2}}} \alpha + {\text{D}}_{{\text{j}}}^{{\text{2}}} \beta + {\text{D}}_{{\text{k}}}^{{\text{2}}} \gamma + {\text{D}}_{{\text{m}}}^{{\text{2}}} \delta } \right)$$

(12)

The two geometrical cases shown in Fig. 9 represent the extreme cases (loosest and densest) of the relative density. Real filters are unlikely to exist either as densest or loosest states, but rather at some intermediate state or relative density. Locke et al. (2001) and Indraratna et al. (2007) proposed that a more realistic pore model should also consider the filter relative density. They suggested that a combination of two cases gives the constriction size d_c, which is computed using the relative density D_r as provided by Eq. 13:

$$d_{c} \left( {P_{c} } \right) = D_{{c3}} \left( {P_{c} } \right) + P_{c} \left( {1 - D_{r} } \right)\left[ {D_{{c4}} \left( {P_{c} } \right) - D_{{c3}} \left( {P_{c} } \right)} \right]$$

(13)

where:

• d_c: the constriction size at the relative density D_r;

• P_c: the probability of occurrence by area of the constriction size d_c.