Abstract

The study of percolation of fluids through permeable subsoil strata has always been a subject of great significance from engineering point of view. The permeability depends upon both material properties and fluid characteristics. Previous studies are concentrated on the influence of regularly shaped particles on permeability, whereas the present study aims to analyse its variation due to natural randomly shaped particles. Seepage velocity of the fluid and hydraulic gradient of the soil bed materials are evaluated with the help of discharge and pressure measurements in a constant head permeameter. Specific gravity tests are conducted on different materials in order to calculate their porosities. The dependence of permeability on Dp/dg (i.e. ratio of permeameter size (Dp) to particle size (dg)) for various wall effect conditions is analysed through standard experimental procedures. The present study examines the effect of relative resistance of permeameter wall (i.e. the ratio of permeability of confined to unconfined bed) on Dp/dg. The results reveal that the aquifer confinement i.e. the distance up to which the region of randomly packing occurs directly influences the measure of permeability at lower values of Dp/dg. This effect decreases with increasing values of Dp/dg and becomes negligible at values 120 and above.

INTRODUCTION

The percolation of fluids through porous media is an important phenomenon that occurs appreciably in many physical situations such as flow through aquifers and in situations where packing material is contained within structures like cooling towers, sewage treatment plants and chemical reactors (Nemec & Levec 2005). Permeability is an essential quantitative measure of fluid transmissibility in the linear, pre-linear and post linear regimes. It depends on the physical properties of flowing liquid such as density and viscosity at the temperature and pressure involved as well as the characteristics of the transmitting medium such as particle size, porosity of the packing, shape of the individual particles, size distribution of the particles in the bed with respect to each other and particle surface roughness and permeameter walls (Bear 2013). The field scenarios in general are represented by sufficiently large porous media in the direction normal to flow, such that the influence of boundary on particles of the medium may be ignored. However, the studies or measurements of permeability in the laboratory suffer with a few constraints due to the extent of medium causing the wall effect to predominate in such situations.

Several theories have been proposed for explaining the phenomena of wall effect on flow through porous media. Cohen & Metzner (1981) are the first to suggest a tri-regional model to facilitate the study of permeability in all three regions separately through detailed description of porosity variations within the region. The model is comprised of a wall region extending up to some distance next to the tube wall, a bulk region along the centre of the tube, and a transition region in between. According to their theory, fluid in each region is assumed to be flowing through parallel capillary tubes. Another alternate model has been proposed by Nield (1983), in which he divided the porous media into two zones viz. a thin layer of the order of one particle diameter adjacent to the wall, which can be safely assumed to be occupied by fluids only, and another layer of a packed bed with uniform porosity whose behaviour can be well described by Darcy's law. The boundary conditions introduced by Beavers & Joseph (1967) are used for the complete analysis of this two-region model. The models described above, however, provided significant insights into the wall effect phenomenon in the earlier times, but the lack of experimental data for values of DP/dg below 6.4 and above 40 limited their applicability for all practical situations.

The study of the influence of wall on fluids flowing through porous media in recent years was also taken up by Allen et al. (2013), Hellström & Lundström (2006), Sahimi (2012), Yadav (2013), as well as others. Their studies deduced that boundary influences flow primarily in two ways: firstly, the velocity at the wall is assumed to be zero to ensure a no-slip condition which introduces a variation in the uniform flow field predicted by Darcy's law (Miguel 2012); and secondly, the packing of particles in an unconsolidated porous medium will change due to the presence of boundary as there will be haphazard packing in its proximity as compared to the compact packing in the central portion (De Klerk 2003). This causes non-uniformity in porosity of the bed, thus affecting its permeability and flow of fluid through it (Larsson et al. 2012). The permeability of the confined bed (K), coefficient of dynamic viscosity, the permeability of the unconfined bed (K**) are some of the factors that are taken into consideration to study the effect of confinement on permeability and are calculated using empirical equations (Choi et al. 2008; Khirevich et al. 2010; Fodor et al. 2011; Hamdan & Kamel 2011). A value of the ratio K/K**, greater than unity, implies that the increase in permeability is mainly due to the effect of increased porosity at the wall region. On the other hand, a ratio of less than unity suggests the role of shearing stresses due to greater surface area at the wall in decreasing the value of permeability (Jin-Sui et al. 2009). It has been observed from previous studies that wall effect becomes negligible for a value of K/K** equal to unity. Furnas (1929) and Cohen & Nield (1985) suggest that surface area is a dominant factor in determining the value of K/K** for a sufficiently thick wall zone. This dependence of permeability of the confined bed on permeameter diameter can be satisfactorily found out by analysing the variation of K/K** values with the ratio DP/dg (Winterberg & Tsotsas 2000). As per Chapuis (2012), the permeability of confined bed and unconfined bed or infinite bed can be determined as:  
formula
where:
  • K = permeability for confined bed (cm/s), K** = permeability for unconfined bed (cm/s)

  • v = velocity of flow or seepage velocity (m/s), i = hydraulic gradient (m/m), υ = kinematic viscosity (cm2/s)

  • g = acceleration due to gravity (m/s2), ɛ = porosity of the bed, given by:

  • Ws = total weight of material contained in permeameter (kg), G = specific gravity of sediment

  • V = volume of permeameter (m3), γw = specific weight of water (N/m3), Z = shape factor of the material, given by: (Z)=(Sp)2/3/Vp, Sp = surface area of the particle (m2) and Vp = the volume of the particle (m3).

Moreover, the shape of the particles of the material occupying the porous bed also affects its resistance and is used as a correction factor for the determination of correct values of permeability from empirical formulae. It can also be calculated by comparing the experimental values of permeability for the given material to the values when particles of the medium are assumed to be spherically shaped. In the present study, shape factor plays an important role due to the usage of natural, randomly shaped materials and is chosen based on an average for a random sample, where the value of the shape factor for a spherical particle can be safely approximated equal to 5 (Loth 2008). The present study for wall effect on fluid flow through porous media has been carried out on three permeameters with pipe diameters (DP) 5.08 cm, 10.16 cm, and 15.24 cm. The objective of conducting these hydraulic tests on materials of various grain sizes is the determination of friction factor (Fr), Reynolds number (Re) and studying the effect of container wall on permeability.

MATERIALS AND METHODS

Hydraulic tests carried out in a laboratory to determine permeability with the experimental setup (ASTM 2011) (Figure 1) comprises of the following:

  • 1.

    Water supply system: An overhead tank situated at a height of about 2.65 m above the permeameter outlet supplies water to the permeameter and receives its own continuous supply from a re-circulating tank so as to maintain a constant head in it.

  • 2.

    Permeameter: A constant head vertical flow type permeameter with three different pipe diameters, viz. 5.08 cm, 10.16 cm, and 15.24 cm, is used for conducting the hydraulic tests. The main permeameter section consists of a tube with an internal diameter of 10.2 cm and a total length of 100 cm. Pressure tapping points at a centre to centre distance of 46 cm in the middle of permeameter tube are provided diametrically opposite to each other at each section. This arrangement of tapping points helps measure mean pressure for every section by averaging out the manometer readings of the tapping points at the corresponding section. The inlet to the permeameter consists of a pipe 19 mm in diameter and the rate of flow of water through the permeameter is regulated with the help of an outlet sluice valve of an equal diameter. A BSS200 mesh screen is also used in the filter for resting the porous media. For filling in and removing the material, permeameter is detached each time from both the top and bottom couplings.

  • 3.

    Manometers: In order to include the desired range of flow, two types of manometers are used, i.e. (1) a U-tube air-water manometer for measurement of head losses of water falling in the range of about 150 cm to 200 cm and (2) a U-tube paraffin-water manometer for differential heads of water of about 20 cm to 0.2 cm.

  • 4.

    Discharge measuring device: In the volumetric method of discharge measurement the water is collected in a bucket for a recorded period of time which is then measured in a 2,000 cm3 graduated jar.

  • 5.

    Packing material: The uniformly sized samples such as that of sand of particle size 0.5 mm; white marble chips with particle sizes 4 mm and 1.18 mm; black marble chips with sizes 0.85 mm, 1.18 mm and 1.40 mm; and gravel with sizes 0.85 mm, 1.40 mm, 1.70 mm and 2.80 mm, are used for filling the permeameter in layers followed by proper compaction.

Figure 1

Experimental setup of the permeability measuring apparatus (ASTM 2011).

Figure 1

Experimental setup of the permeability measuring apparatus (ASTM 2011).

In a hydraulic test, pressure drop is measured across the test length of the material. The experimental investigation involved use of sieve analysis for sorting uniform sized material and to determine the geometric mean diameter from the pore size of the sieves. The specific gravity studies are carried out by pycnometer method to determine the porosities of the bed materials. The temperature of water was recorded at the beginning and end of each run to ensure that the value of Reynolds number did not change due to change in the viscosity with changing temperature.

RESULTS AND DISCUSSION

The experimental investigations are carried out to study the relation of permeability with the properties of the material and the medium. Different ranges of parameters and the material characteristics are used for obtaining required inputs for study.

Variation of Fr vs. Re for different mean size

Uniformly sized white and black marble chips, gravel and sand particles with their mean grain diameter ranging between 0.5 mm to 4.0 mm are used for different sets of experiments. The relationships between the dimensionless quantities Fr (Huang et al. 2013 and Re are shown in Figures 2 and 3), where:  
formula
and  
formula
Figures 2 and 3 indicate straight line behaviour for flow up to the critical Re value signifying a linear regime of flow as the viscous forces predominate over the inertial and other forces and also the validity of Darcy's law (Alabi 2011). In general, the value of critical Re varies between 2 and 5 but, in the present study, Re obtained a critical value of 10. This is due to the involvement of various factors such as the shape, size, grading, roughness, packing and porosity of the materials used, which greatly influences the properties of the porous media (Wu & Yu 2007). The porosity of bed for different material used is observed to vary between the values of 0.346 and 0.371, which is a small range of variation.
Figure 2

Plot between Fr and Re for white and black marble chips of different sizes.

Figure 2

Plot between Fr and Re for white and black marble chips of different sizes.

Figure 3

Plot between Fr and Re for sand and gravel of different sizes.

Figure 3

Plot between Fr and Re for sand and gravel of different sizes.

The wall effect

The results of the experiments conducted in order to study the behaviour of variations in permeability with changing values of DP/dg ratio are shown in Figure 4. The results from the Chu & Nag (1989) model have also been depicted in the same figure. Figure 4 allows the measured response to be analysed with the help of polynomial equations:  
formula
for proposed model and  
formula
for the Chu & Nag (1989) model, where y is the corresponding response (dependent variable), x is independent variable and R2 is coefficient of determination. A close comparison of the two curves led to the following conclusions:
Figure 4

Plot between K/K** and Dp/dg for the present and Chu & Nag's (1989) model.

Figure 4

Plot between K/K** and Dp/dg for the present and Chu & Nag's (1989) model.

  • 1.

    The Chu & Nag (1989) model can only be used for the determination of permeability for values of DP/dg less than 40.

  • 2.

    The value of K/K** is greater than unity for DP/dg less than 40 from both the models.

  • 3.

    The value of K/K** decreases continuously for the ratio of DP/dg lying between 40 and 60 and is approximately unity for a value DP/dg more than 120, in the present model.

From the above conclusions, it is safely inferred that for values of DP/dg less than 40, i.e. for thinner beds, higher porosity in the wall (or boundary) zone due to haphazard packing, dominates over the influence of higher surface area per unit volume, but the observed phenomena for the value of DP/dg lying between 40 and up to the value of 120 is a complete reversal of the former. These experimental results are also in good agreement with the data reported by Chu & Nag (1989) and Cohen & Nield (1985) models, with the values of K/K** lying on a slightly higher scale as compared to those reported by the Chu & Nag (1989) model. The exaggeration in the values of K/K** is a result of the presence of a greater number of flow channels in natural material due to their irregular shape or flakiness leading to a higher porosity as compared to that of standard spherical material. For values of DP/dg greater than 120, no wall effect is observed to affect the values of permeability as the permeameter boundary can now be assumed to be as good as infinite.

The experimental results for the natural material utilized for the present study, along with their shape factors, has been tabulated in precise numerical values in Table 1 and the results as depicted in Figure 4 are observed to be in good agreement with the data reported by various researchers in their previous studies. Table 1 shows the values of particle size, the size of the permeameter used, shape factor, DP/dg ratio, porosity, permeability for finite bed and the ratio K/K**, in that order.

Table 1

Experimentally determined values of K/K** for different materials

S. no. Material dg (mm) Dp(mm) Z Dp/dg ɛ K (cm/s) K/K** 
1. White marble chips 4.00 50.8 7.60 12.700 0.3710 11.6013 1.5926 
2. Gravel 2.80 50.8 6.90 18.1400 0.3640 5.6771 1.4186 
3. Gravel 1.70 50.8 6.10 29.8800 0.3610 2.3705 1.3002 
4. Gravel 1.40 50.8 6.20 36.2857 0.3512 1.1249 1.0764 
5. Gravel 0.85 50.8 6.20 59.7647 0.3485 0.3499 0.9374 
6. Black marble chips 1.40 101.6 7.67 72.5700 0.3480 0.6465 0.9827 
7. Black marble chips 1.18 101.6 7.62 86.1020 0.3472 0.4652 1.0001 
8. Black marble chips 0.85 50.8 7.59 59.764 0.3468 0.1277 0.9544 
9. White marble 1.18 152.4 7.49 129.15 0.3507 0.4964 0.9967 
10. Sand 0.50 101.6 6.10 119.5294 0.3460 0.2507 1.0362 
S. no. Material dg (mm) Dp(mm) Z Dp/dg ɛ K (cm/s) K/K** 
1. White marble chips 4.00 50.8 7.60 12.700 0.3710 11.6013 1.5926 
2. Gravel 2.80 50.8 6.90 18.1400 0.3640 5.6771 1.4186 
3. Gravel 1.70 50.8 6.10 29.8800 0.3610 2.3705 1.3002 
4. Gravel 1.40 50.8 6.20 36.2857 0.3512 1.1249 1.0764 
5. Gravel 0.85 50.8 6.20 59.7647 0.3485 0.3499 0.9374 
6. Black marble chips 1.40 101.6 7.67 72.5700 0.3480 0.6465 0.9827 
7. Black marble chips 1.18 101.6 7.62 86.1020 0.3472 0.4652 1.0001 
8. Black marble chips 0.85 50.8 7.59 59.764 0.3468 0.1277 0.9544 
9. White marble 1.18 152.4 7.49 129.15 0.3507 0.4964 0.9967 
10. Sand 0.50 101.6 6.10 119.5294 0.3460 0.2507 1.0362 

CONCLUSIONS

The present study reveals that the upper limit of the validity of the seepage flow depends primarily on the shape, size, grading, roughness and packing of beds. The permeameter wall plays a vital role in the determination of permeability only at low values of DP/dg and becomes negligible for values of DP/dg greater than 120. For the estimation of correct values of permeability, the shape factor of the particles should be given due consideration for flow through natural materials, as both the permeability and the wall effect are equally dependent on the changes in porosity introduced on varying the particle shapes. The proposed model for estimating permeability provides accurate influence of wall effect, which has impact on the correct assessment of seepage. On the basis of the experimental investigations conducted, it is reported that the apparent permeability increases with decreasing ratio of DP/dg up to a value less than 50, which is also in agreement with the data reported by Furnas (1929) and Cohen & Nield (1985). Whereas the apparent permeability of the infinite bed (K**) is almost equal to that of the finite bed (K) for values of DP/dg between 40 and 80 and thereafter, any changes in the value of permeability are only due to factors other than the boundary, as mentioned above. The study recommends further research on effect of grain shape and wall surface roughness on permeability.

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