Determination of hydraulic conductivity for granular filters based on constriction size and shape parameters

Granular filters are important structural components of hydraulic structures provided with an aim to prevent the migration of fine particles of base soil and at the same time allow the free movement of seepage flow in order to avoid the pore pressures. Hydraulic conductivity denotes the ease with which flow takes place through any porous medium (Terzaghi 1922; Jabro 1992). The importance of hydraulic conductivity for the design of filters has been discussed by many researchers (Bertram 1940; US Army Corps 1953; Sherard et al. 1984; Kenney et al. 1985; USBR 1987; Indraratna et al. 1996; Alsakran & Zhu 2020). Boadu (2000) concluded that the measurement of hydraulic conductivity by laboratory and field methods was costly and time consuming, and insufficient field data of aquifers and hydraulic boundaries add to the limitation (Freeze & Cherry 1979; Todd & Mays 2005). These limitations have led researchers to shift to analytical approaches for the study of groundwater and the porous media (Salmasi & Azamathulla 2013; Cheng & Hsu 2021). Hazen (1893), Carman (1939), Sherard et al. (1984), Vukovic & Soro (1992), Indraratna et al. (1996) and Ishaku et al. (2011) conducted investigations to suggest empirical equations for the hydraulic conductivity determination based on particle sizes of granular soils. Kenney et al. (1984), Odong (2007) and Trani & Indraratna (2010) assessed void network of the granular medium as an important parameter for hydraulic conductivity compared to the particle size. Indraratna et al. (2007) highlighted the advantages of applying the method of constriction size for granular soils. The relationship between constriction size and hydraulic conductivity of granular soils was investigated by Indraratna et al. (2012) and Seblany (2018). Also, studies, although limited have been conducted on the influence of shape and gradation parameters on filter permeability (Rather et al. 2020). Several studies have been carried out for determination of hydraulic conductivity of granular soils based on particle size, shape parameters as well as constriction size of void network. It was concluded from the literature that there was less focus on the combined effect of constriction size and shape parameters on the hydraulic conductivity of granular filters.

The objective of the present study was to determine hydraulic conductivity of granular material using constriction size of the void network and the shape parameters of the filter material. Laboratory tests were carried out to investigate their relationship; for this purpose five shapes of granular filter material were used; further nine gradations for each shape were studied to deduce the relationship between these parameters. Experimental data was further analysed to arrive at an empirical model for prediction of hydraulic conductivity that considers constriction size of void network and shape parameters of filter material.

In the present study, hydraulic conductivity of filter material was determined using two important parameters i.e. constriction size and shape parameters. Constriction size (d*) is the smallest opening between the void spaces of filter material. This size is responsible for controlling the migration of fine particles of base material and determines the value of hydraulic conductivity. Incorporation of constriction size in the hydraulic conductivity computations refines the accuracy of results. Many theoretical concepts have been suggested to work out the constriction size from the particle gradation (Lone et al. 2005, Indraratna & Raut 2006, Indraratna et al. 2007; Raut & Indraratna 2008; Indraratna et al. 2015). In the present work, the constriction size of void network of filter material was determined by the procedure given by Lone et al. (2005). A unit assembly of three non-uniform spheres was considered and these spheres were assumed to be in contact with each other in a manner such that their centres were in a single plane forming a window whose sides are of concave arcs with radii equal to the radii of the assembly spheres. The arrangement of unit assembly is shown in Figure 1.

The m, n, and curves given by Lone et al. (2005) can be used to determine the values of ⁠. The above discussed model has been used to determine the constriction size of filter mass by grouping the entire filter material into different skeleton particle sizes.

The material used for experimentation and test procedure of the experimental runs for the present study are discussed in this section.

The granular filter material required for the laboratory investigations was obtained from a local river bed and grouped into different standard sizes ranging from 4.75 to 50 mm on the basis of particle diameter. Further, the material was sorted into five shapes, designated as S₁, S₂, S₃, S₄ and S₅ based on their shape parameters. The shape parameters considered in the present study are sphericity, shape factor, elongation ratio, and flatness ratio; designated as Sph, SF, ER and FR respectively. Nine filter material gradations (M, M₂, M₃, M₄, M₅, M₆, M₇, M₈ and M₉) of each shape were prepared for laboratory investigations. The shape parameters of the filter material used in the present study are given in Table 1.

The constant head permeability apparatus was used to determine the hydraulic conductivity of filter material (ASTM D2434-68 2006) as shown in Figure 2. The test apparatus consists of a cylinder having diameter of 250 mm, length of 600 mm with an 80 mm diameter of hopper base. The movement of filter material in the steel cylinder due to the inflowing water was prevented by providing the wire mesh of different openings ranging from 2.5 to 8 mm. At the top plate of the cylinder, a pressure gauge and two air vents were provided. A stopcock was also provided to regulate the water supply at the inlet. The outflow from the apparatus was allowed into the measuring tank with the help of a flexible pipe of reasonable height for measuring the discharge of water required for the test. Metric staffs were provided for the measurement of head.

In Equation (2), the coefficient of permeability and the viscosity of water at 20 °C and T°C are denoted by k₂₀, k_t, μ₂₀ and μ_t respectively.

The prepared samples of filter material were tested and the laboratory results are given in Table 2. The experimental results clearly reveal the variation of hydraulic conductivity with the shape of filter material even for similar gradations. The relationship of hydraulic conductivity of filter material and constriction size for the five shapes are given in Figure 3(a)–3(e). The corresponding empirical models are given in Table 3.

The experimental results obtained in this study indicate that the hydraulic conductivity of filter material having similar constriction sizes differs with the filter particle shape. Hence, it is necessary to incorporate the shape parameters for prediction of hydraulic conductivity of filter material based on constriction size. The relationship of hydraulic conductivity of filter material with constriction size of void network and individual shape parameters are given in Figure 4(a)–4(e).

The proposed empirical models for prediction of hydraulic conductivity based on controlling constriction size and individual shape parameters are given in Table 4.

The experimental results reveal that constriction size of void network and shape parameters of filter particles have significant effect on hydraulic conductivity of filter material. The relationship of hydraulic conductivity of filter material with constriction size of void network and shape parameters (sphericity, flatness ratio, shape factor and elongation ratio) is given in Figure 5.

The model proposed for determining hydraulic conductivity of filter material was validated by selecting permeability data, shape parameters and constriction size of filter material used in the research work carried out by Lone et al. (2005). The computed hydraulic conductivity values, shape parameters and constriction size of filter material used in the study by Lone et al. (2005) and the predicted values of hydraulic conductivity by the proposed model Equation (3) are given in Table 5. The values of hydraulic conductivity of filter material computed by Lone et al. (2005) were in close approximation to the values of hydraulic conductivity of filter material predicted by Equation (3). The observed versus predicted hydraulic conductivity of filter material is presented in Figure 6.

The present work was carried out with an objective to determine the relationship of hydraulic conductivity with constriction size and shape parameters of filter material. The main findings of the study are summarized as follows:

• Hydraulic conductivity of filter material was found to be the function of controlling constriction size and shape parameters.

• Empirical relations were developed for hydraulic conductivity determination of filters based on constriction size of different filter particle shapes.

• An empirical model based on power regression of significant parameters like constriction size and the shape parameters of filter material was developed for the computation of hydraulic conductivity.

The prediction of hydraulic conductivity is a pre-requisite for the design of structures that encounter seepage flow such as hydraulic structures constructed on permeable soils. To ensure the safety of such structures, protective filters form an integral component of the set-up. The design of protective filters depends upon the hydraulic conductivity of filter material and the base material. The model proposed in the present study is more suitable for the prediction of hydraulic conductivity as it encompasses shape as well as the constriction size of the filter assembly. It is very clear from the results of this study that the shape of the filter particles and the constriction size have considerable effects on the hydraulic conductivity; therefore, their incorporation in the model for hydraulic conductivity determination will improve the accuracy of estimation.

The aim of the present work was to incorporate effect of shape into the empirical models for hydraulic conductivity determination. Because of the time limit of the study conducted and the exhaustive procedure of sorting the shapes, only five shapes of filter particles could be studied. This being one of the limitations of the study, it is highly recommended to extend this study for more shapes so as to improve the efficacy of the design criteria for filters. Also, there is scope of improving the outcome, if the number of shape parameters considered are increased and their determination is conducted in the laboratory by latest material testing techniques.

This paper presents the research work carried out by the author*, who received a PhD fellowship from the Ministry of Human Resources Development (MHRD), Government of India. The experimental work was carried out in the Fluid Mechanics Laboratory, NIT Srinagar, Jammu and Kashmir, India.

No potential conflict of interest was reported by the authors.

The data used for the development of empirical models in the present study are given in Table 2 in the Results and discussion section. The data used for the validation of proposed empirical model are given in Table 5 of the validation section.

All relevant data are included in the paper or its Supplementary Information.