Transport experiments were performed in a horizontal column under constant flow conditions using the step-input injection method. In this investigation, a Plexiglas column with an inner diameter of 6.4 cm and a length of 33 cm is used. The column was equipped with four equally spaced (9 cm) piezometers, allowing one to measure pressure variations during the suspended particles’ injection in the porous medium. The column was packed in 5 cm increments by pouring the sand into the column, mixed with deionised water (saturated conditions) to avoid trapping air bubbles. Using a Master-Flex peristaltic pump (Cole-Parmer Instruments), the column is fed by two reservoirs (Fig. 1); the first one contains deionised water (pH of 6.8 ± 0.1) and the other one contains suspended particles subjected to a permanent agitation with the help of a motorised stirrer. When the steady-state flow conditions are reached with deionised water, the valve is switched to the second reservoir and the flow of suspended particles is directed toward the column. So, a perfect step-input injection was assumed to be achieved, even if Taylor dispersion occurs at the boundary of the system (Mainhagu et al., 2012). However, the distance between the reservoir and the cell was kept as short as possible in order to minimize the Taylor dispersion effect.

The detection system consists of a Kobold Instrument turbidimeter (calibrated with respect to the suspended particles concentrations) and a Turner Designs 10-AU fluorometer (calibrated with respect to the fluorescein concentrations).

The used material as porous medium is alluvial sand (collected from Sebaou River, Algeria), whose grains present a smooth angular and elongated shape. Two size distributions were selected from this material (Fig. 2): fine medium, and coarse medium. A third medium (noted mixture) was obtained by mixing the fine and the coarse media (50% mass from each). The shape of the grain size distributions of the used media are linked to the uniformity coefficient C_u, whose values are reported in Table 1. Coarse and fine media provide a narrowly graded curve, while the mixture presents a broadly graded size distribution.

Permeability tests were carried out to characterize the porous media (clean bed). The column was fed from the deionised water reservoir under constant head. The measurements of the different pressures along the column allowed us to determine the permeability of the porous medium using Darcy's law. The porosity of various media was measured using the bulk density method. Table 1 summarizes the main characteristics of the used porous media.

A mercury porosimeter (Micromeritics) was used to measure the pore size distribution of the three media. The results show that the mixture presents the widest pore size distribution (Fig. 3) with two modal pore size values (200 μm and 450 μm), while coarse and fine media provide narrow distributions with modal pore size close to 550 μm and 280 μm, respectively. The average pore diameters are 336.6 μm, 237.3 μm and 182.2 μm respectively for coarse, fine and mixture media. These results show that the mixture of the coarse and fine sands leads to build narrow pores, owing to the small grains that take place between the coarse ones. Studies conducted with several porous media show that the porosity and pore size decrease for a wide grain size distribution (Bradford and Bettahar, 2006; Porubcan and Xu, 2011). X–ray synchrotron microtomography study of the assemblage of silt grains with a clay phase showed that the voids resulting can be considered as fully connected (Rozenbaum et al., 2012).

The injected fine particles are of quartz silt selected from Sebaou River alluviums. According to the grain size analysis (Malvern Multisizer 2000), the particle size ranges from 0.3 to 40 μm, with an average diameter d_p50 = 7 μm. The suspended particles and dissolved tracer (fluorescein) injection experiments were conducted with a constant injected concentration C₀ = 0.5 g/L and 100 μg/L, respectively. Eight tracer tests were performed either with dissolved tracer or suspended particles for each medium at flow rates ranging from 55 to 750 mL/min, corresponding to Darcy's velocities U ranging between 0.029 and 0.39 cm/s. The Reynolds number is defined by Re = Ud_g50ρ/ν, where d_g50 is the mean size grain of the medium, ρ is the fluid density, and ν is the dynamic viscosity. The values of Re obtained over the tested porous media range between 0.32 and 9.75, which confirms that the experiments were conducted under laminar flow conditions (de Marsily, 1986). Thus, Darcy's law remains applicable in this investigation. The molecular Péclet number is defined as P_ed = U_pd_g50/D₀, where U_p is the pore velocity and D₀ is the molecular diffusion coefficient. For D₀ = 10⁻⁹ m²/s (diffusion of fluorescein in water), P_ed values range between 800 and 9600. Also, these experiments conform to regime IV (300 < P_ed < 10⁵) defined by Pfannkuch (1963) for the dissolved tracer, indicating that mechanical dispersion is the main dominant process and that molecular diffusion can be neglected.

To determine the size of particles transported out of the porous medium, the samples were collected for grain size analysis using a laser device (Malvern Multisizer, 2000).

With the assumption that the deposition of the suspended particles does not substantially modify the properties of the porous medium for low concentrations and a short duration of test (measures being analyzed over 3.5 pore volumes), and under steady-state and saturated flow conditions, the transport of suspended particles can be described by the convection–dispersion equation with a first order deposition kinetics (Kretzschmar et al., 1997; Wang et al., 2000):

The initial and boundary conditions for a semi-infinite medium are given in Eq. (2):

The analytical solution of equation (1) according to equation (2) is as follows (Van Genuchten, 1981):

Eq. (3) can be written as follows:

When K_dep = 0 (φ = 1), Eq. (6) becomes the classic one for a dissolved tracer case (Ogata and Banks, 1961). Sauty (1977) showed graphically that the second term of equation (6) can be neglected when the dynamic Péclet number (Pe_dyn) is larger than 20. Over all the tests performed, the calculated values of Pe_dyn range between 19 and 58, and the calculations show that the amount representing the second term of the analytical solution remains less than 5% of simplified solution. Then, it can be assumed that the second term of equation (6) can be neglected, and so equation (6) reduces to:

When time tends toward ∞, the concentration reaches the plateau C_p given by equation (8):

Dividing equation (7) by equation (8), we obtain:

Drawing on the linear graphical method (Wang et al., 1987), the above simplification leading to the previous equation allows the transformation of equation (9) as follows:

By matching the measured breakthrough curves (BTCs) with the provided analytical solution, pore velocity U_p, longitudinal dispersion coefficient D_L and kinetic deposition coefficient K_dep were determined. The filtration coefficient λ was so deduced by the following equation (Kretzschmar et al., 1997):

The transformation of the analytical solution [Eq. (10)] by the linear regression method (LRM) provides a practical way to identify the model parameters. The best fit of our experimental curves with the analytical solution is obtained for both dissolved tracer and suspended particles with different tested flow velocities (Fig. 4). The dissolved tracer is completely recovered (C/C₀ = 1) whatever the flow velocity. However, the suspended particles recovery varies with the flow velocity and its values are less than 85% (i.e. C/C₀ < 0.85), indicating that a portion of the particles is retained in the porous medium. The higher the flow velocity is, the higher the maximum concentration and the shorter the residence time are. The recovery rate is obtained by integrating the experimental breakthrough curve over the test duration, and the results show that it increases with the flow velocity, as reported in the literature (Benamar et al., 2005; Grolimund et al., 1998; Kretzschmar et al., 1997). However, the recovery rate obtained for the particles is low (ranging from 32% for the lowest flow velocity to 86% for the highest one), which indicates a systematic capture of the coarsest fractions: particles larger than 25 μm are not recovered (see further, Section 3.2.2).

The recovery mass is obtained through the integration of the breakthrough curve over time experiment. Fig. 5 presents the variation of the recovery rate R of the silt particles with the pore velocity (U_p) for the considered porous media. The general trend is a non-linear (logarithmic) increase of R with pore velocity. The recovery rate observed for the particles at the lowest velocity was close to 30%, while the highest velocity provides a recovery rate greater than 80%, indicating that more particles are moved by the flow from the soil to the outlet. When comparing the recovery rate within the three porous media, the results show that the recovery decreases when the average pore size decreases. The mixture medium provides the lowest recovery rate (highest retention rate) owing to the wide distribution of pore size, forming multiple retention sites (sites of constriction, crevice, and cavern). Also, weak pore connectivity involves actively the capture of suspended particles (Elimelech and O’Melia, 1990; Lahav and Tropp, 1980; Wang et al., 2000).

When the values of the diffusion Péclet number are larger than 300, the molecular diffusion may be neglected (Pfannkuch, 1963), and the hydrodynamic dispersion can be expressed by Eq. (12):

Hu and Brusseau., 1994 suggested that the value of coefficient m ranges between 1.0 and 1.3. According to Ahfir et al. (2009), the values of m range rather between 0.75 and 0.90.

The coefficient of longitudinal dispersion D_L was determined for both used tracers (dissolved tracer and suspended particles). Fig. 6 displays the variation of the dispersion coefficient of suspended particles with the pore velocity within the three tested media. The results show that the longitudinal dispersion coefficient increases with increasing pore water velocity. The same evolution was observed for the different tested media. The relationship between the dispersion coefficient and the pore velocity was rather described by a linear law, and the longitudinal dispersivity value, α_L, was close to 0.8 cm. According to equation (12), the coefficient m for these tests takes a value close to unit. Thus, for the two types of tracers (dissolved and particles), the variation of the dispersion coefficients with pore velocity is shown in Fig. 7 and the longitudinal dispersivity α_L of each tracer was derived. When comparing the dispersion of the two tracers within the tested porous media, the dispersion coefficient of the suspended particles is greater than that of the dissolved tracer, and the difference increases with increasing flow velocity. The results further demonstrate that the dispersivity of the dissolved tracer (α_L = 0.66) is smaller than that of the suspended particles (α_L = 0.80 cm) over the media investigated (Fig. 7). This behaviour can be explained by the large size distribution of the suspended particles (0.3 to 40 μm), which involves a large dispersion of the flow velocities of the particles. The difference behaviour, which is more marked for large flow velocities (Ahfir et al., 2009) owing to the inertial effects, can also be linked to the size exclusion effect, which is more marked at high flow velocities (Benamar et al., 2007; Kretzschmar et al., 1997). The use of an experimental device (Hele–Shaw cell) to investigate the dispersion process inside a fracture (Mainhagu et al., 2012) showed that the longitudinal macro-dispersion obtained suggests an asymptotical behaviour of the plume spread. The longitudinal dispersivity derived from moments method was found to be increasing linearly with the Péclet number computed at the mass centre of the plume.

This section discusses the comparison between the transport behaviour of the suspended particles and the dissolved tracer. The results reported in literature (Ahfir et al., 2007; Benamar et al., 2005; Grolimund et al., 1998; Kretzschmar et al., 1997; Masséi et al., 2002) reveal that the breakthrough of the suspended solid particles occurred earlier than the dissolved tracer, and suggest that the fast transport of the suspended particles compared to the dissolved tracer is attributed to the size exclusion effect where the suspended particles are transported in preferential pathways. Corapcioglu and Jiang (1993), Grindrod et al. (1994) have shown that the transfer time of the colloidal particles is lower than that of the solute molecules, which can be accessed by their small size and their high diffusion coefficient to an important share of the porosity space. However, results reported by Masséi et al. (2002) show that the longer transfer time of suspended particles could occur beyond a threshold flow velocity value.

The residence time is defined as the time when a half of the maximum concentration is reached in the breakthrough curve. In this study, the residence time of the suspended particles in the three tested media is longer than that of the dissolved conservative tracer. The retardation factor, noted V_r, is defined as the ratio between the flow velocities of particles and the dissolved tracer (i.e. the ratio between the dissolved tracer's residence time and the suspended particles’ residence time). Fig. 8 displays the variation of the retardation factor V_r as a function of the pore flow velocity U_p. Whatever the flow velocity and the porous medium tested, V_r is always lower than 1. Thus, the residence time of the particles in the medium is longer than that of the dissolved tracer. However, the recovery delay of the suspended particles decreases when flow velocity increases and the retardation factor approaches the unit value in the mixture medium for the highest velocity tested (Fig. 8). The lower the flow velocity is, the more significant is this retardation of particles in comparison with the dissolved tracer. Such a result is in accordance with those obtained by Masséi et al. (2002) with similar flow velocities. According to Fig. 8, the retardation of the recovered particles is generally greater in the coarse medium than in the fine one, the lowest retardation being observed in the mixture medium. This behaviour is related to the availability of trapping in narrow constrictions and settling of heavy particles. Due to the large porosity of the coarse medium and the low pore velocity, the particles are trapped or evolve by collisions with grains, leading to a recovery delay. An enhanced collision effect would affect particles, whereas a dissolved tracer would be rather unconcerned by such mechanical effects. Further results about recovered particles size evidenced that larger particles are provided by the coarse medium, suggesting that the delay of restored particles increases with their size. This is likely due to the gravitational and inertial effect. When the inertia of a particle is high, it may no longer follow exactly the streamlines and will therefore be able to impact the medium's grains. The greater the inertia of a particle is, the greater is the probability of collisions between suspended particles and grains. The effect of gravity increases with particle size and can lead to a more complex trajectory than that of the dissolved tracer.

The deposition kinetics coefficient K_dep is determined from breakthrough curves adjustment. The best fit values of K_dep range from 9 to 36 h⁻¹. The lower values are associated with the coarse medium and the lowest velocity, while the greater values are observed for the highest velocity within the mixture medium. The deposition kinetics coefficient is linked to the filtration coefficient through Eq. (10). The proportionality between the deposition kinetics coefficient and the pore velocity leads to a unique coefficient λ that can be used to evaluate the filtration behaviour. The filtration coefficient λ, for the three tested media, decreases with increasing flow velocity (Fig. 9), according to an exponential trend. The comparison of the filtration coefficient values from different media shows that the mixture medium provides the highest values, while the coarse medium presents the lowest ones. These results are consistent with those widely reported in literature (Grolimund et al., 1998; Masséi et al., 2002; Porubcan and Xu, 2011; Xu and Saiers, 2009). By using the microscopic approach (Djehiche et al., 2009), the collector–particle model showed that the collector efficiency decreases as the flow's velocity increases.

When the mass-averaged grain and particle sizes (respectively d_g50 and d_p50 for which 50% of the mass is finer) were used to represent the size of such materials, there appeared to be an increase in coefficient K_dep with ratio d_p50/d_g50 for the uniform media (coarse and fine), while the mixture medium presents a greater value of K_dep even if the value of the ratio d_p50/d_g50 locates between the values of both uniform media. This trend is similar to that obtained for the dispersivity and reveals that the representation of a medium by the averaged size parameters may lead to some discrepancy when heterogeneous materials are investigated. If considering the results obtained by Porubcan and Xu (2011) for colloids retention in fine sands, the lowest K_dep value obtained in this study is many times higher than that obtained by previous authors for colloids. In accordance with their results, when using the mass-averaged grain to represent the size of the porous media, the variation of K_dep with the ratio d_p50/d_g50 shows that the values of K_dep obtained using the mixture medium were higher than those estimated from our experiments with the two other media.

3.2.1 Particle size distribution during breakthrough

In this section, the analysis of the particle size distribution of the suspended particles recovered at the column outlet is presented. Effluent samples were collected at three periodic times during the recovery process of suspended particles in order to carry out a particle size analysis (Malvern Multisizer 2000). Sampling times T₁, T₂, and T₃ correspond to a number of injected pore volume equal to 1, 1.3, and 3 V_p, respectively (Fig. 4). The mass-averaged size d_p50 being used to represent the mean size of the recovered particles, Fig. 10 presents the evolution of the average diameter of the recovered particles from the experiments carried out with different flow velocities in the fine medium. The average size of the suspended particles initially injected is close to 7 μm. For clarity, only the size analysis performed for some flow velocities are presented. The particle size distributions show that among the recovered particles, the coarser ones (10–25 μm) are transferred more rapidly than the fine ones. The same result was obtained by Masséi et al. (2002) using a pulse injection of the suspended particles. The finest particles arrive later compared to the larger ones, and the particle average size decreases with decreasing the flow's velocity (Fig. 10). The maximum concentration during recovery (time T₃) involves a great part of the fine particles. Large particles are excluded from the section of the pores that are smaller than the diameter of the particles. The average diameter d_p50 decreases over time during the breakthrough, rapidly from time T₁ to time T₂, and then slightly to time T₃. This result can be related to the filtration of the coarse particles occurring after one pore volume (T₂ to T₃) of inlet. The finest particles, recovered later, can access a large part of the pore structure and are delayed with respect to the coarsest particles. Most of them spread more widely in the medium and are delayed because of a more complex trajectory than that of coarser particles that take preferential paths consisting of large pores, owing to size exclusion phenomena (Masséi et al., 2002).

3.2.2 Effect of flow velocity and pore structure on particles recovery (size selection)

It is usual in soil filter design to address the filter criteria through the opening filter size and the maximum particle size. In order to assess the suitability of such criterion of mechanical straining, the ratios d_g50/d_p50 and d_g50/d_p90 of the three tested media were derived. The ratios values obtained are respectively (357, 157 and 228.5) and (125, 55 and 80) for coarse, fine and mixture media. If considering the ratio 200 as an upper limit for particle blocking, the mechanical straining increases from fine medium to mixture medium, and finally to coarse medium. Fig. 11 displays the variation of the mass-averaged size of the recovered particles with the flow velocity for the three media. The average size of the recovered particles increases with increasing the flow velocity and the relationship is represented by a logarithmic trend for the three media. The average diameter of recovered particles increases from mixture to fine medium and then to coarse medium. The smaller the pores size (from coarse medium to mixture medium), the finer are the recovered particles.