Hydraulic conductivity (K) estimation of porous media is of great significance in contaminant movement and groundwater investigations. The present study examines the influence of effective grain size (d10) and standard deviation (σ) on the K value of borehole soil samples using 5.08, 10.16, and 15.24 cm diameter permeameters. A statistical grain size model was developed and the feasibility of seven empirical equations was evaluated with the measured K values. The K of soil samples increases with the increase in the d10 grain size and decreases with the increase in the σ value. Evaluation of K using empirical equations establishes that the Hazen equation shows relatively good agreement with the measured K values. The study substantiates the efficacy of the developed model as the Kmodel and Kmeasured based R2 (determination coefficient), MAE (mean absolute error), and RMSE (root mean square error) values are (0.982, 0.007, and 0.008), (0.972, 0.005, and 0.007), (0.953, 0.004, and 0.005) for 5.08, 10.16, and 15.24 cm diameter permeameters respectively. The developed model was validated by assessing its efficiency in the prediction of K values for independent soil samples. The developed model-based K accedes to the precise computation of the aquifer yield and groundwater recharge.

  • The study proposes a statistical grain size model for the computation of hydraulic conductivity of porous media by investigating the influence of the σ/d10 parameter on hydraulic conductivity.

  • The developed hydraulic conductivity model provides an efficient tool to compute the aquifer yield, groundwater recharge, and filter design with precise accuracy.

Graphical Abstract

Graphical Abstract
Graphical Abstract

Understanding the concept of hydraulic conductivity (K) is a fundamental objective for hydrologists in groundwater and geotechnical investigations, and management practices (Pliakas & Petalas 2011; Pucko & Verbovsek 2015). The concept of K was first postulated by Henry Darcy and defined as the ease with which the fluid flow can take place through the interconnected voids (Wang et al. 2017). Darcy's equation correlates the fluid velocity with the hydraulic gradient. The proportionality constant in that equation is termed as the hydraulic conductivity of porous media (Kasenow 2002; Ghanbarian et al. 2016). The computation of dimensionless fluid quantities namely friction factor and Reynolds number of porous media is important to govern the flow regime (Li et al. 2019). The K of the porous media is dependent invariably on the fluid properties i.e., viscosity and specific weight, and the characteristics of the porous media which include the particle size, structural configuration, grain size distribution, and compaction of soil particles (Zieba 2017; Chandel et al. 2021). The investigations regarding the prediction of K of porous media are important to determine the transmitting capacity of the soil particles or the bed, as these factors are dependent on the aquifer recharge or extraction (Cronican & Gribb 2004; Zhu et al. 2021).

For computing the K of porous media, there exist several methods namely field and laboratory methods, and empirical equations (Alabi 2011). The field methods involve cost factors and long testing time which makes them a less reliable approach. The laboratory methods also sometimes prove to be uneconomical due to time restrictions and cost (Boadu 2000). In addition to this, it is challenging to collect the porous media samples representing the actual field conditions for experimental investigations and for field methods, the unavailability of the precise knowledge of the boundary and the aquifer geometry can be a restrictive factor (Riha et al. 2018). However, the prediction of K using empirical equations is a reliable approach for the rough computation of K via grain size parameters because the data related to the textural characteristics of the soil particle are effortlessly and quickly obtained (Rosas et al. 2014; Chandel & Shankar 2021). Moreover, numerous groundwater researchers have focused to establish a relationship between K and grain size parameters, as such proposed relationships are not dependent on the aquifer boundaries and relatively less expansive as compared to other techniques (Lu et al. 2012; Cabalar & Akbulut 2016).

Various researchers in the past such as Vienken & Dietrich (2011), and Banerjee et al. (2019) examined different modelling techniques for computing the K value of porous media. These studies presented elucidations having some limitations i.e, simulating the infinite extent of subsurface flow (Zahiri & Najafzadeh 2018). The one crucial limitation is to control other parameters that affect the K value of porous media while investigating the influence of one particular parameter i.e., effective and mean size (d10 and d50), standard deviation about the mean, and dispersion measure (d50–d10) (Boadu 2000; Song et al. 2009). Krumbein & Monk (1943) stated that the K can be expressed as the product of the exponential and power function of the standard deviation and mean size respectively. Alyamani & Sen (1993) investigated the influence of the grain size parameter on the K of porous media. From this study, it was postulated that the finer grain zone within the grain size curve is vital in the prediction of hydraulic conductivity. Barr (2001) proposed an equation based on Darcy's concept to predict the K of porous sediments. The proposed equation is composed of the parameters which can be computed directly, rather than assuming i.e., shape factor, fluid viscosity, and porosity. Odong (2007) and Ishaku et al. (2011) examined the K of porous media using various grain size-based empirical equations and concluded that the empirical equations should be applied within particular domains of applicability. Pliakas & Petalas (2011) examined the correlation between the statistical grain size parameters with the computed K values. Salarashayeri & Siosemarde (2012) developed an equation based on multiple regression analysis using grain diameters i.e., d10, d50, and d60, and postulated that grain diameter (d10) proved to be an effective parameter in computing the K of porous media. Pucko & Verbovsek (2015) compared different techniques such as empirical grain size and pumping methods to estimate the K of porous media. The study concluded that the grain size methods result in the precise prediction of K compared to the field techniques. Naeej et al. (2017) proposed an equation to estimate the K of porous media using M5 model regression analysis. The M5 model is the binary decision tree, which includes the linear regression equations. Further, the K values from the proposed equation were evaluated with the values calculated from empirical equations and concluded that the developed equation performs better in computing the K of porous media. Wang et al. (2017) proposed a new equation based on the gradation characteristics namely effective grain size (d10), porosity, and uniformity coefficient to predict the K of porous media. Arshad et al. (2020) concluded that the effective grain size (d10) substantially gives better results in the K prediction using empirical models as compared to the mean grain size (d50). Existing literature establishes that various researchers correlated the K with various influencing parameters instead of the standard deviation and effective grain size of porous media. The effective grain size represents the entire characteristics of the grain size curve, whereas the standard deviation measures the precise scatter of grain size values from the mean particle size. Moreover, in the previous studies, the developed equations to predict the K are valid for a small range of grain size parameters.

Based on the above discussion, the present study emphasizes the effect of grain size distribution, which has a substantial influence on the K of the porous media. The two parameters namely effective grain size (d10) and standard deviation (σ) have been integrated in the investigation, which are easily computed for any porous media and has, significant impact on the behavior of grain size distribution. The primary objectives of the study are:

  • (1)

    To study the variation between the Friction factor (Fr) and Reynolds number (Re) for different sizes of porous media to govern the flow regime.

  • (2)

    To study the influence of standard deviation (σ) and effective grain size (d10) on experimentally measured K of porous media.

  • (3)

    To develop a model for computing the K of porous media, involving grain size parameters (σ and d10), and assess the efficacy of the developed model vis-a-vis the pre-existing empirical equations based on the comparative evaluation of the computed and measured K values.

  • (4)

    To validate the developed model for computing the K for a vide range of porous media.

Materials

In the present work, 27 representative borehole soil samples were collected from the Kangra district of Himachal Pradesh in India. The soil samples were acquired from an ongoing drilling operation, which was established to locate the aquifer geological profile. In order to obtain the undisturbed soil samples, thin-walled sampler tubes having a diameter and length of 8.5 and 115 cm respectively were used, so that the collected samples are subjected to minimum disturbance concerning the actual field conditions. The study area has a ground elevation of 425–3,500 m above the mean sea level and lies in the north-west region of the Himalayas. The latitude and longitude of the study area are 31°3′N and 75°25′E respectively. The soil characteristics of the study area comprise sand, gravel, and silts. During the drilling operation, three soil samples from nine boreholes were collected, which were drilled at an interval of 5–6 m apart. The soil samples were collected at an interval of 3 m from the core material and then subjected to dry sieve analysis in the laboratory for further experimental investigations.

Methodology

Initially, the collected soil samples were subjected to dry sieve analysis using a mechanical sieve shaker device to determine the effective and mean size of each sample. The specific gravity was computed using the standard pycnometer method, which is imperative for the porosity determination. The effective size d10 is specified by standard guidelines using sieve analysis and the value of standard deviation is computed from the relation (Todd & Mays 2005) as:
(1)
where, = standard deviation, d50 = grain size corresponding to 50% finer by weight, di = particle diameter retained on a specific sieve, and ΔZi = fraction of total particles.

The resistance offered to the fluid flow through porous media was assessed by conducting the hydraulic test. The K of soil samples was estimated using a Constant Head Permeameter (CHP) having internal diameter i.e., 5.08, 10.16, and 15.24 cm as shown in Figure 1.

Figure 1

Experimental setup of K measuring apparatus.

Figure 1

Experimental setup of K measuring apparatus.

Close modal
Figure 2

Gradation curve of soil samples 1–15.

Figure 2

Gradation curve of soil samples 1–15.

Close modal
For estimating the K value, Figure 1 represents the line diagram of the experimental setup, which is established in the laboratory. The setup comprises permeameters which are manufactured from Galvanized iron pipe having a total and test length of 106 cm and 46.5 cm respectively. An overhead tank is located at a height of 2.65 m above the permeameter outlet. The overhead tank supplies the water to the permeameter and receives water supply from the re-circulating tank continuously to maintain a constant head level. Pressure taping points are provided along the periphery of the permeameter at a center to center distance of 46.5 cm to measure the pressure difference readings. The arrangement of pressure taping points helps in recording the manometer readings. The permeameter consists of an inlet and outlet pipe of 19 mm diameter each to regulate the water flow rate through the permeameter. The discharge measurement is done by collecting the water in a measuring cylinder for an appropriate time interval i.e., 30 sec via a digital stopwatch. For one sample 5–6 discharge readings have been recorded and the average of the discharge value is used to compute the hydraulic conductivity. The K value of soil samples was measured via the standard procedure as described by Rosas et al. (2014) and ASTM (2006). The water temperature was measured using a digital thermometer at the start and end of each analysis. The hydraulic conductivity of soil samples has been determined using Darcy's equation (Qiu & Wang 2015):
(2)
where, Q = flow volume (m3/s), L = test length (m), h = head difference between the pressure taps (m), and A = sample cross-sectional area (m2).

In this study, the influence of d10 and on the K of 15 borehole soil samples was investigated individually. Further, the behaviour of hydraulic conductivity, obtained from three diameter permeameters have been analyzed with the different values of , which provides the basis for developing a statistical grain size model using the principle of least squares approach. For the model development, a data set of 45 hydraulic conductivity values have been used in the study.

Empirical equations for K estimation

The prediction of K using empirical equations depends primarily on the viscosity, uniformity coefficient, grain size, porosity, and sorting coefficient. Numerous researchers have performed experimental investigations to develop the empirical equations by analyzing the interrelationship between these parameters. Vukovic & Soro (1992) and Song et al. (2009) have given a generalized hydraulic conductivity equation for various empirical equations:
(3)
where, K = hydraulic conductivity (m/s), = dimensionless coefficient, depends on the various porous media parameters (grain size and shape, structure, and heterogeneity) ν = kinematic viscosity (m2/s); g = acceleration due to gravity (m/s2), f(n) = porosity function, represents the degree of porous media compactness, and de = effective diameter (m).

Various researchers have proposed different empirical equations by analyzing the relationships of K with various influencing parameters i.e., de, f(n), and ν. Based on Equation (3), the empirical equations have been reoriented and written in the standard form. Table 1 shows the empirical equations used in this study for the hydraulic conductivity evaluation with their applicability limits. Also, the two recently proposed empirical equations namely Chapuis et al. (2005) and Naeej et al. (2017) have been incorporated for the hydraulic conductivity assessment. The g and ν values in the empirical equations considered in this study were taken as 981 cm/s2 and 0.885 mm2/s respectively.

Table 1

Empirical equations for K prediction

ResearcherEquationApplicability limits
Hazen (1892)   0.1 mm < d10 < 3 mm, U < 5 
Slichter (1899)   0.01 mm < d10 < 5 mm 
Terzaghi (1925)   Large grain sand 
Kozeny–Carman (Kozeny 1927; Carman 1937, 1956 Suitable for gravel and sand, d10 < 3.0 mm 
Harleman et al. (1963)   Coarse and well grained media 
Chapuis et al. (2005)   0.03 mm ≤ d10 ≤ 3 mm 
Naeej et al. (2017)   Valid for sand and gravel < 3.0 mm 
ResearcherEquationApplicability limits
Hazen (1892)   0.1 mm < d10 < 3 mm, U < 5 
Slichter (1899)   0.01 mm < d10 < 5 mm 
Terzaghi (1925)   Large grain sand 
Kozeny–Carman (Kozeny 1927; Carman 1937, 1956 Suitable for gravel and sand, d10 < 3.0 mm 
Harleman et al. (1963)   Coarse and well grained media 
Chapuis et al. (2005)   0.03 mm ≤ d10 ≤ 3 mm 
Naeej et al. (2017)   Valid for sand and gravel < 3.0 mm 

Where, n = porosity, U = uniformity coefficient i.e., d60/d10.

Statistical performance indicators

For quantitative assessment between the measured and computed K values, various statistical indicators i.e., BIAS, scatter index (Si), determination coefficient (R2), agreement index (Ia), mean absolute error (MAE), and root mean square error (RMSE) have been used in this study (Naeej et al. 2017). The statistical indicators are defined as:
(4)
(5)
(6)
(7)
(8)
(9)
where, Z is the number of datasets, and denote the measured and computed K values respectively. and represent the average values of measured and computed parameters, respectively.

Experimental investigations include the grain size analysis and CHP test, which have been performed on the collected borehole soil samples. A total number of 27 soil samples have been used i.e., 15 soil samples to study the influence of σ and d10 on K and for model development whereas, the remaining 12 soil samples were considered as independent samples for the validation of the developed model.

Grain-size analysis

Initially, the grain size analysis was conducted on the collected borehole soil samples and then the grain size curve for the 15 soil samples was plotted between the percent finer and particle size as shown in Figure 2. From the grain size curve the grain sizes (d10, d30, d50, and d60), uniformity coefficient, and standard deviation values were determined. For the remaining soil samples, the grain size parameters (σ and d10) have been determined and mentioned in the validation section.

Figure 3

Fr and Re variation for different soil samples from (a) 5.08, (b) 10.16, and (c) 15.24 cm diameter permeameters.

Figure 3

Fr and Re variation for different soil samples from (a) 5.08, (b) 10.16, and (c) 15.24 cm diameter permeameters.

Close modal

The basic properties of the 15 soil samples is presented in Table 2. The values of d10 and vary between 0.173–0.386 mm and 1.470–7.100 respectively.

Table 2

Basic properties of the soil samples

Sample no.Gravel (%)Sand (%)Silt (%)d10 (mm)d30 (mm)d50 (mm)d60 (mm)naUaσa
25.78 68.76 5.46 0.386 0.821 1.450 1.850 0.359 4.793 1.470 
27.41 70.46 2.13 0.373 0.774 1.330 1.720 0.363 4.611 1.560 
31.24 66.78 1.98 0.358 0.690 1.180 1.530 0.370 4.274 1.640 
21.91 74.82 3.27 0.347 0.631 1.050 1.400 0.375 4.035 1.790 
18.73 76.84 4.43 0.342 0.575 0.980 1.240 0.385 3.626 1.980 
16.28 81.46 2.26 0.330 0.544 0.920 1.160 0.387 3.515 2.190 
12.64 83.71 3.65 0.325 0.499 0.850 1.120 0.389 3.446 2.460 
14.82 81.73 3.45 0.301 0.486 0.785 1.070 0.386 3.555 2.780 
19.43 76.17 4.40 0.297 0.468 0.720 0.962 0.394 3.239 2.980 
10 11.84 85.29 2.87 0.278 0.442 0.650 0.929 0.392 3.342 3.320 
11 7.54 89.98 2.48 0.225 0.346 0.570 0.754 0.392 3.351 3.780 
12 10.51 85.92 3.57 0.191 0.300 0.490 0.590 0.398 3.089 4.130 
13 8.94 88.19 2.87 0.187 0.264 0.410 0.497 0.410 2.658 4.890 
14 10.73 85.34 3.93 0.178 0.260 0.370 0.425 0.418 2.388 5.870 
15 13.93 83.13 2.94 0.173 0.252 0.335 0.372 0.426 2.150 7.100 
Sample no.Gravel (%)Sand (%)Silt (%)d10 (mm)d30 (mm)d50 (mm)d60 (mm)naUaσa
25.78 68.76 5.46 0.386 0.821 1.450 1.850 0.359 4.793 1.470 
27.41 70.46 2.13 0.373 0.774 1.330 1.720 0.363 4.611 1.560 
31.24 66.78 1.98 0.358 0.690 1.180 1.530 0.370 4.274 1.640 
21.91 74.82 3.27 0.347 0.631 1.050 1.400 0.375 4.035 1.790 
18.73 76.84 4.43 0.342 0.575 0.980 1.240 0.385 3.626 1.980 
16.28 81.46 2.26 0.330 0.544 0.920 1.160 0.387 3.515 2.190 
12.64 83.71 3.65 0.325 0.499 0.850 1.120 0.389 3.446 2.460 
14.82 81.73 3.45 0.301 0.486 0.785 1.070 0.386 3.555 2.780 
19.43 76.17 4.40 0.297 0.468 0.720 0.962 0.394 3.239 2.980 
10 11.84 85.29 2.87 0.278 0.442 0.650 0.929 0.392 3.342 3.320 
11 7.54 89.98 2.48 0.225 0.346 0.570 0.754 0.392 3.351 3.780 
12 10.51 85.92 3.57 0.191 0.300 0.490 0.590 0.398 3.089 4.130 
13 8.94 88.19 2.87 0.187 0.264 0.410 0.497 0.410 2.658 4.890 
14 10.73 85.34 3.93 0.178 0.260 0.370 0.425 0.418 2.388 5.870 
15 13.93 83.13 2.94 0.173 0.252 0.335 0.372 0.426 2.150 7.100 

aRepresents the unitless parameters.

Variation between friction factor and Reynolds number

The dimensionless flow parameters i.e., Fr and Re were computed for soil samples using permeameter with different diameters i.e., 5.08, 10.16, and 15.24 cm. A logarithmic plot has been drawn between the dimensionless flow parameters as shown in Figure 3. The computation of Fr and Re values plays a crucial role in determining the flow regime. The Fr and Re are computed as:
(10)
(11)
where, hi = hydraulic gradient, V = average flow velocity, d50 = mean grain size, and υ = fluid kinematic viscosity, and g = gravitational constant.

Figure 3 indicates straight-line variation between Fr and Re for permeameters having different diameters, which signifies that the flow is in a linear regime and thereby confirms the presence of Darcy's regime (Hellstrom & Lundstrom 2006; Alabi 2011).

Variation of K with effective grain size and standard deviation

The hydraulic conductivity values obtained from the three permeameters were plotted against the effective grain size i.e., d10 (Figure 4(a)), wherein a linear variation of K was observed with the different values of d10. From this investigation, it is concluded that as the value of d10 increases, it results in providing more void space to the fluid to move through the interconnected voids, and thereby results in an increase in the K value which is in the line with the outcome of Pliakas & Petalas (2011). The K value varies from 0.342 to 0.068 cm/s, 0.271 to 0.054 cm/s, and 0.232 to 0.048 cm/s for 5.08, 10.16, and 15.24 cm diameter permeameter respectively.

Figure 4

Variations of K with (a) effective grain size; and (b) standard deviation.

Figure 4

Variations of K with (a) effective grain size; and (b) standard deviation.

Close modal

Figure 4(b) represents the variation of K with the values of standard deviation, which vary from (1.47–7.10) for 5.08, 10.16, and 15.24 cm diameter permeameters. The hydraulic conductivity value decreases with the increase in the value. The observed value of for the soil samples is greater than 1, which represents the non-uniformity of porous media. As the value of σ increases the non-uniformity of porous particles increases which impart more compactness to the packed media and thereby results in the decreased value of the hydraulic conductivity. As the K value decreases and tends to zero, the curves approach asymptotically towards the y-axis as shown in Figure 4(b).

Hydraulic conductivity variation with σ/d10

The variation of hydraulic conductivity computed from three permeameters was studied by plotting K with different values of σ/d10. Figure 5 indicates that as the σ/d10 value increases K decreases. The curve touches asymptotically the lower limiting value and is concave upward. The inferences were drawn from the study, i.e., the trend of the curve in Figure 5 is in close agreement with the outcomes of Masch & Denny (1966) and Pliakas & Petalas (2011). The σ/d10 values range between 3.80 and 41.06.

Figure 5

Variations of K with σ/d10.

Figure 5

Variations of K with σ/d10.

Close modal

Investigations on the influence of σ/d10 on the hydraulic conductivity values obtained from different diameter permeameters provide the basis for developing a novel statistical model for the computation of K of porous media.

Statistical model development

A statistical model for the computation of hydraulic conductivity has been proposed by using the data points of 15 borehole soil samples. The grain size parameters i.e., σ and d10 have been used for model development. The developed statistical model comprises coefficients of σ/d10 parameter of degree 0–4. The proposed model is developed using the principle of least squares analysis.

The standard form of the proposed K model is given below:
(12)
where, = factor that considers the compactness of the soil particles near the wall, particle roughness, wall confinement, and extent of the porous media. The values of λ for different diameter permeameters are given below:
Permeameter diameter (cm)λ values
5.08 1.23 
10.16 0.96 
15.24 0.82 
Permeameter diameter (cm)λ values
5.08 1.23 
10.16 0.96 
15.24 0.82 

From the observed values of λ, it can be seen that the λ value is larger for minimum permeameter diameter and smaller for maximum permeameter diameter. This observation postulates that the magnitude of λ becomes insignificant with the increase in the porous media extent.

The empirical constant (z0, z1, z2, z3, and z4) values are: z0 = 0.3880, z1 = 0.0327, z2 = 0.0013, z3 = 2.56 × 10−5, and z4 = 2 × 10−7.

The developed statistical model for the K estimation is:
(13)

Further, the σ/d10 and K values computed experimentally and by using the developed statistical model are given in Table 3.

Table 3

Experimentally measured (Kmeaured) and developed model (Kmodel)-based K values

Sample no.σ/d10(Kmeaured) (cm/s) permeameter diameter (cm)
(Kmodel) (cm/s) permeameter diameter (cm)
5.0810.1615.245.0810.1615.24
3.805 0.342 0.271 0.232 0.346 0.270 0.231 
4.390 0.323 0.249 0.219 0.330 0.257 0.220 
5.539 0.298 0.240 0.200 0.300 0.234 0.200 
6.669 0.272 0.215 0.188 0.273 0.213 0.182 
7.876 0.246 0.197 0.169 0.248 0.194 0.165 
9.058 0.221 0.181 0.150 0.226 0.176 0.151 
10.787 0.190 0.156 0.132 0.199 0.155 0.132 
13.255 0.162 0.131 0.109 0.168 0.131 0.112 
15.104 0.145 0.121 0.097 0.150 0.117 0.100 
10 17.376 0.126 0.102 0.088 0.133 0.104 0.089 
11 21.754 0.109 0.090 0.072 0.114 0.089 0.076 
12 25.964 0.098 0.079 0.065 0.106 0.083 0.071 
13 29.868 0.088 0.068 0.059 0.104 0.082 0.070 
14 34.467 0.078 0.061 0.053 0.093 0.076 0.064 
15 41.066 0.069 0.054 0.048 0.082 0.065 0.052 
Sample no.σ/d10(Kmeaured) (cm/s) permeameter diameter (cm)
(Kmodel) (cm/s) permeameter diameter (cm)
5.0810.1615.245.0810.1615.24
3.805 0.342 0.271 0.232 0.346 0.270 0.231 
4.390 0.323 0.249 0.219 0.330 0.257 0.220 
5.539 0.298 0.240 0.200 0.300 0.234 0.200 
6.669 0.272 0.215 0.188 0.273 0.213 0.182 
7.876 0.246 0.197 0.169 0.248 0.194 0.165 
9.058 0.221 0.181 0.150 0.226 0.176 0.151 
10.787 0.190 0.156 0.132 0.199 0.155 0.132 
13.255 0.162 0.131 0.109 0.168 0.131 0.112 
15.104 0.145 0.121 0.097 0.150 0.117 0.100 
10 17.376 0.126 0.102 0.088 0.133 0.104 0.089 
11 21.754 0.109 0.090 0.072 0.114 0.089 0.076 
12 25.964 0.098 0.079 0.065 0.106 0.083 0.071 
13 29.868 0.088 0.068 0.059 0.104 0.082 0.070 
14 34.467 0.078 0.061 0.053 0.093 0.076 0.064 
15 41.066 0.069 0.054 0.048 0.082 0.065 0.052 

Computation of K using empirical equations

The hydraulic conductivity of soil samples was computed via the seven empirical equations considered in this study, as mentioned in Table 1. Grain-size parameters, uniformity and sorting coefficients, and porosity values were used to compute K based on empirical equations. The value of grain size parameters is stated in Table 2. A specific value of the sorting coefficient has been used in each empirical equation except the Terzaghi equation. In the Terzaghi's equation two different values of sorting coefficient i.e., 10.7 × 10−3 for smooth grains and 6.1 × 10−3 for coarse grains have been given, therefore an average sorting coefficient value of 8.4 × 10−3 has been used in the Terzaghi equation (Pucko & Verbovsek 2015). The computed values of hydraulic conductivity via empirical equation are shown in Table 4.

Table 4

Hydraulic conductivity values computed using empirical equations

Sample no.KHazen (cm/s)KSlichter (cm/s)KTerzaghi (cm/s)KK−C (cm/s)KHarleman et al. (cm/s)Kchapuis et al. (cm/s)KNaeej et al. (cm/s)
0.197 0.057 0.098 0.154 0.108 0.057 0.039 
0.188 0.055 0.095 0.151 0.101 0.056 0.039 
0.179 0.054 0.093 0.150 0.093 0.056 0.039 
0.172 0.053 0.092 0.150 0.087 0.056 0.039 
0.175 0.056 0.098 0.162 0.085 0.060 0.041 
0.164 0.053 0.093 0.155 0.079 0.058 0.041 
0.161 0.053 0.092 0.153 0.077 0.057 0.041 
0.136 0.044 0.076 0.127 0.066 0.049 0.037 
0.137 0.046 0.080 0.135 0.064 0.052 0.039 
10 0.119 0.039 0.069 0.116 0.056 0.046 0.036 
11 0.078 0.026 0.045 0.076 0.037 0.033 0.030 
12 0.058 0.020 0.034 0.058 0.026 0.027 0.028 
13 0.058 0.021 0.036 0.064 0.025 0.029 0.030 
14 0.054 0.020 0.035 0.063 0.023 0.028 0.030 
15 0.053 0.020 0.035 0.065 0.022 0.029 0.031 
Sample no.KHazen (cm/s)KSlichter (cm/s)KTerzaghi (cm/s)KK−C (cm/s)KHarleman et al. (cm/s)Kchapuis et al. (cm/s)KNaeej et al. (cm/s)
0.197 0.057 0.098 0.154 0.108 0.057 0.039 
0.188 0.055 0.095 0.151 0.101 0.056 0.039 
0.179 0.054 0.093 0.150 0.093 0.056 0.039 
0.172 0.053 0.092 0.150 0.087 0.056 0.039 
0.175 0.056 0.098 0.162 0.085 0.060 0.041 
0.164 0.053 0.093 0.155 0.079 0.058 0.041 
0.161 0.053 0.092 0.153 0.077 0.057 0.041 
0.136 0.044 0.076 0.127 0.066 0.049 0.037 
0.137 0.046 0.080 0.135 0.064 0.052 0.039 
10 0.119 0.039 0.069 0.116 0.056 0.046 0.036 
11 0.078 0.026 0.045 0.076 0.037 0.033 0.030 
12 0.058 0.020 0.034 0.058 0.026 0.027 0.028 
13 0.058 0.021 0.036 0.064 0.025 0.029 0.030 
14 0.054 0.020 0.035 0.063 0.023 0.028 0.030 
15 0.053 0.020 0.035 0.065 0.022 0.029 0.031 

Further, the K values determined using different diameter permeameters were compared with the values computed via the empirical equations considered in the study. The Hazen equation gives a substantially closer agreement with the measured K values for all soil samples, followed by the Kozeny–Carman equation as shown in Figure 6. The Hazen equation depends on the entire particle distribution curve, porosity, and effective particle size which makes this equation more precise in the computation of K as compared to the other equations (Carrier 2003; Ishaku et al. 2011). Whereas the other empirical equations i.e., Slichter (1899), Terzaghi (1925), Harleman et al. (1963), Chapuis et al. (2005), and Naeej et al. (2017), resulted in poor agreement with the experimentally measured K values, which is consistent with the findings of Cheng & Chen (2007) and Riha et al. (2018) as indicated in Figure 6. The Terzaghi equation results in lower K values because of using the average value of sorting coefficient, whereas, in Chapuis et al. (2005) equation, the parameters namely fluid viscosity and gravitational acceleration are not considered as compared to other equations, which, may result in the underestimation of K values. Slichter (1899), Harleman et al. (1963), and Naeej et al. (2017) underestimated the K values because these empirical equations and for a well graded porous media (Rosas et al. 2014).

Figure 6

Comparison of measured and computed K for (a) 5.08, (b) 10.16, and (c) 15.24 cm diameter permeameters.

Figure 6

Comparison of measured and computed K for (a) 5.08, (b) 10.16, and (c) 15.24 cm diameter permeameters.

Close modal

Further, the hydraulic conductivity values computed using the developed statistical model have been compared with the experimentally measured K values as shown in Figure 7. The K values obtained from the developed model show a relatively good agreement with the measured K values as compared to the empirical equations considered in the study. Figure 7 shows that the data points computed from the developed model are focused more on the agreement line as compared to Figure 6.

Figure 7

Comparison of measured and developed model-based K values.

Figure 7

Comparison of measured and developed model-based K values.

Close modal

Further, the quantitative performance of the developed model and empirical equations were assessed using different statistical indicators for different diameter permeameters as given in Table 5.

Table 5

Statistical indicators for developed model and empirical equations

Permeameter diameterStatistical parametersDeveloped modelHazenSlichterTerzaghiKozeny–CarmanHarleman et al.Chapuis et al.Naeej et al.
5.08 cm BIAS 0.007 −0.056 −0.143 −0.113 −0.066 −0.121 −0.138 −0.148 
Si 0.045 0.390 0.883 0.717 0.483 0.740 0.865 0.932 
R2 0.982 0.874 0.794 0.783 0.824 0.802 0.722 0.593 
Ia 0.984 0.951 0.881 0.891 0.915 0.879 0.883 0.861 
MAE 0.007 0.056 0.143 0.113 0.066 0.121 0.138 0.148 
RMSE 0.008 0.072 0.163 0.132 0.089 0.136 0.160 0.172 
10.16 cm BIAS 0.002 −0.019 −0.107 −0.076 −0.029 −0.084 −0.101 −0.112 
Si 0.047 0.227 0.822 0.616 0.343 0.643 0.800 0.884 
R2 0.972 0.893 0.819 0.808 0.842 0.834 0.749 0.621 
Ia 0.985 0.946 0.904 0.875 0.938 0.896 0.889 0.863 
MAE 0.005 0.025 0.107 0.076 0.034 0.084 0.101 0.112 
RMSE 0.007 0.034 0.121 0.091 0.051 0.095 0.118 0.130 
15.24 cm BIAS 0.002 0.003 −0.084 −0.054 −0.007 −0.062 −0.079 −0.089 
Si 0.039 0.177 0.776 0.537 0.276 0.565 0.751 0.849 
R2 0.953 0.879 0.802 0.791 0.846 0.865 0.731 0.604 
Ia 0.978 0.953 0.902 0.885 0.938 0.913 0.876 0.862 
MAE 0.004 0.018 0.084 0.054 0.026 0.062 0.079 0.089 
RMSE 0.005 0.022 0.097 0.067 0.035 0.071 0.094 0.106 
Permeameter diameterStatistical parametersDeveloped modelHazenSlichterTerzaghiKozeny–CarmanHarleman et al.Chapuis et al.Naeej et al.
5.08 cm BIAS 0.007 −0.056 −0.143 −0.113 −0.066 −0.121 −0.138 −0.148 
Si 0.045 0.390 0.883 0.717 0.483 0.740 0.865 0.932 
R2 0.982 0.874 0.794 0.783 0.824 0.802 0.722 0.593 
Ia 0.984 0.951 0.881 0.891 0.915 0.879 0.883 0.861 
MAE 0.007 0.056 0.143 0.113 0.066 0.121 0.138 0.148 
RMSE 0.008 0.072 0.163 0.132 0.089 0.136 0.160 0.172 
10.16 cm BIAS 0.002 −0.019 −0.107 −0.076 −0.029 −0.084 −0.101 −0.112 
Si 0.047 0.227 0.822 0.616 0.343 0.643 0.800 0.884 
R2 0.972 0.893 0.819 0.808 0.842 0.834 0.749 0.621 
Ia 0.985 0.946 0.904 0.875 0.938 0.896 0.889 0.863 
MAE 0.005 0.025 0.107 0.076 0.034 0.084 0.101 0.112 
RMSE 0.007 0.034 0.121 0.091 0.051 0.095 0.118 0.130 
15.24 cm BIAS 0.002 0.003 −0.084 −0.054 −0.007 −0.062 −0.079 −0.089 
Si 0.039 0.177 0.776 0.537 0.276 0.565 0.751 0.849 
R2 0.953 0.879 0.802 0.791 0.846 0.865 0.731 0.604 
Ia 0.978 0.953 0.902 0.885 0.938 0.913 0.876 0.862 
MAE 0.004 0.018 0.084 0.054 0.026 0.062 0.079 0.089 
RMSE 0.005 0.022 0.097 0.067 0.035 0.071 0.094 0.106 

The values of Ia and R2 vary from 0 to 1, and Si, MAE, RMSE, and BIAS from 0 to ∞ (Naeej et al. 2017). BIAS may result in negative values which indicates the lower values of computed parameters as compared to the measured parameter. Values closer to 1 for Ia and R2, and lower values of Si, MAE, RMSE, and BIAS indicate a better correlation between measured and computed parameters. Among the seven empirical equations considered in the study, the Hazen equation results in the lower values of Si, MAE, RMSE, and BIAS, and values are close to 1 for Ia and R2 for different diameter permeameters as shown in Table 5. This postulates that the Hazen equation performs relatively well in K estimation as compared to the other equations. The values of statistical indicators i.e., Si, MAE, RMSE, and BIAS are 0.045, 0.007, 0.008, & 0.007 for 5.08 cm, 0.047, 0.005, 0.007, and 0.002 for 10.16 cm, and 0.039, 0.004, 0.005, and 0.002 for 15.24 cm diameter permeameters respectively are significantly reduced for the developed statistical model, whereas the Ia and R2 values are 0.984 and 0.982, 0.985 and 0.972, and 0.978 and 0.953 for 5.08, 10.16, and 15.24 cm diameter permeameters respectively. The quantitative evaluation via the statistical indicators establishes the efficacy of the developed statistical model in computing the K of porous media.

Validation of the developed model

For the validation of the developed model, the data points corresponding to the remaining 12 soil samples have been used. Initially, the standard deviation and effective grain size for these samples have been determined. The developed model is composed of the parameter ‘σ/d10’ for computing the K value. Therefore, the values of σ/d10 for these soil samples have been determined as shown in Table 6.

Table 6

Hydraulic conductivity predicted using the developed model

Sample no.Gravel (%)Sand (%)Silt (%)σ/d10(Kpredicted) (cm/s)
(Kmeasured) (cm/s)
Permeameter diameter (cm)
Permeameter diameter (cm)
5.0810.1615.245.0810.1615.24
22.86 74.46 2.68 3.519 0.355 0.277 0.236 0.364 0.268 0.227 
20.37 75.68 3.95 3.831 0.345 0.270 0.230 0.325 0.255 0.222 
21.65 76.49 1.86 4.800 0.319 0.249 0.212 0.314 0.244 0.208 
18.62 75.94 5.44 5.714 0.295 0.231 0.197 0.275 0.204 0.227 
16.76 79.74 3.50 4.930 0.315 0.246 0.210 0.321 0.252 0.202 
14.81 82.85 2.34 4.215 0.335 0.261 0.223 0.312 0.228 0.260 
15.69 81.76 2.55 5.818 0.293 0.229 0.195 0.248 0.185 0.168 
13.14 84.26 2.60 6.920 0.268 0.209 0.178 0.275 0.215 0.180 
12.18 86.79 1.03 7.982 0.246 0.192 0.164 0.205 0.168 0.138 
10 11.76 85.46 2.78 10.792 0.199 0.155 0.132 0.158 0.145 0.109 
11 9.86 88.57 1.57 15.850 0.144 0.112 0.096 0.118 0.082 0.069 
12 10.39 86.75 2.86 20.160 0.119 0.093 0.080 0.091 0.078 0.056 
Sample no.Gravel (%)Sand (%)Silt (%)σ/d10(Kpredicted) (cm/s)
(Kmeasured) (cm/s)
Permeameter diameter (cm)
Permeameter diameter (cm)
5.0810.1615.245.0810.1615.24
22.86 74.46 2.68 3.519 0.355 0.277 0.236 0.364 0.268 0.227 
20.37 75.68 3.95 3.831 0.345 0.270 0.230 0.325 0.255 0.222 
21.65 76.49 1.86 4.800 0.319 0.249 0.212 0.314 0.244 0.208 
18.62 75.94 5.44 5.714 0.295 0.231 0.197 0.275 0.204 0.227 
16.76 79.74 3.50 4.930 0.315 0.246 0.210 0.321 0.252 0.202 
14.81 82.85 2.34 4.215 0.335 0.261 0.223 0.312 0.228 0.260 
15.69 81.76 2.55 5.818 0.293 0.229 0.195 0.248 0.185 0.168 
13.14 84.26 2.60 6.920 0.268 0.209 0.178 0.275 0.215 0.180 
12.18 86.79 1.03 7.982 0.246 0.192 0.164 0.205 0.168 0.138 
10 11.76 85.46 2.78 10.792 0.199 0.155 0.132 0.158 0.145 0.109 
11 9.86 88.57 1.57 15.850 0.144 0.112 0.096 0.118 0.082 0.069 
12 10.39 86.75 2.86 20.160 0.119 0.093 0.080 0.091 0.078 0.056 

By using the σ/d10 value for different soil samples and λ value for different diameter permeameters, the K value has been determined using the developed model. Further, for validation, the predicted values of K have been compared with the values measured using the permeameters for these 12 soil samples as shown in Figure 8.

Figure 8

Comparison of predicted K with the measured values.

Figure 8

Comparison of predicted K with the measured values.

Close modal

From Figure 8 it has been observed that the hydraulic conductivity values predicted using the developed model shows fairly good agreement with the measured K values. During validation of the developed model the values of statistical indicators i.e., R2, Ia, BIAS, Si, MAE, and RMSE are 0.942, 0.894, 0.044, 0.052, 0.025, and 0.018 respectively, which substantiate the performance of the developed model in computing the K value of porous media.

The present study is devoted to establishing a statistical model for the computation of the hydraulic conductivity based on the grain size parameters i.e., d10 and σ. The developed model includes a factor ‘λ’ which incorporates the porous media compactness, particle roughness, and extent of the porous media. The observed values of λ indicate that the magnitude of λ becomes insignificant with the increase in the porous media extent. The influence of d10 and σ on the K of borehole soil samples elucidate that, with the increase in the d10 grain size, the K of soil samples increases and decreases with the increase in the σ value. The Fr and Re variations indicate the existence of flow in Darcy's regime. The comparative evaluation of seven empirical equations indicates that the Hazen equation shows a relatively good agreement with the measured K values. The quantitative evaluation using the statistical indicators substantiates the efficacy of the developed statistical model in the computation of K of porous media. The BIAS, Si, R2, Ia, MAE, and RMSE for the developed model are 0.007, 0.045, 0.982, 0.984, 0.007, and 0.008 for 5.08 cm, 0.002, 0.047, 0.972, 0.985, 0.005, and 0.007 for 10.16 cm, 0.002, 0.039, 0.953, 0.978, 0.004, and 0.005 for 15.26 cm diameter permeameters respectively. The study also validates the performance of the developed model in computing the K for the independent data set. The developed model provides an effective tool to compute the aquifer yield, groundwater recharge, and filter design with precise accuracy.

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

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