Prediction of hydraulic conductivity of porous media using a statistical grain-size model

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 R 2 (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.


GRAPHICAL ABSTRACT INTRODUCTION
Understanding the concept of hydraulic conductivity (K) is a fundamental objective for hydrologists in groundwater & 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 & 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;. 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 & 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 is effortlessly and quickly obtained (Rosas et al. 2014;. Moreover, numerous groundwater researchers focus 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 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 of porous media while investigating the influence of one particular parameter i.e., effective and mean size (d 10 & d 50 ), standard deviation about the mean, and dispersion measure (d 50 -d 10 ) (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., d 10 , d 50 , and d 60 , and postulated that grain diameter (d 10 ) 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 (d 10 ), porosity, and uniformity coefficient to predict the K of porous media. Arshad et al. (2020) concluded that the effective grain-size (d 10 ) substantially gives better results in the K prediction using empirical models as compared to the mean grain-size (d 50 ). 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 (d 10 ) 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 & 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, twenty-seven 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 0 N and 75°25 0 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 d 10 is specified by standard guidelines using sieve analysis and the value of standard deviation is computed from the relation (Todd & Mays 2005) as: where, s ¼ standard deviation, d 50 ¼ grain-size corresponding to 50% finer by weight, d i ¼ particle diameter retained on a specific sieve, and ΔZ i ¼ 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, & 15.24 cm as shown in Figure 1.
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).
where, Q ¼ flow volume (m 3 /s), L ¼ test length (m), h ¼ head difference between the pressure taps (m), and A ¼ sample crosssectional area (m 2 ). In this study, the influence of d 10 and s 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 s=d 10 , 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: where, K ¼ hydraulic conductivity (m/s), a ¼ dimensionless coefficient, depends on the various porous media parameters (grain size and shape, structure, and heterogeneity) ν ¼ kinematic viscosity (m 2 /s); g ¼ acceleration due to gravity (m/s 2 ), f(n) ¼ porosity function, represents the degree of porous media compactness, and d e ¼ effective diameter (m). Various researchers have proposed different empirical equations by analyzing the relationships of K with various influencing parameters i.e., d e , f(n), and ν. Based on Equation (3), the empirical equations have been reoriented and written in the  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/s 2 and 0.885 mm 2 /s respectively.

Statistical performance indicators
For quantitative assessment between the measured and computed K values, various statistical indicators i.e., BIAS, scatter index (S i ), determination coefficient (R 2 ), agreement index (I a ), 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: where, Z is the number of datasets, K i & X i denote the measured and computed K values respectively. K & X represent the average values of measured and computed parameters, respectively. Large grain sand Kozeny-Carman (Kozeny 1927;Carman 1937Carman , 1956)

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 3. From the grain-size curve the grain sizes (d 10 , d 30 , d 50 , & d 60 ), uniformity coefficient, and standard deviation values were determined. For the remaining soil samples, the grain-size parameters (σ & d 10 ) have been determined and mentioned in the validation section. The basic properties of the 15 soil samples have been presented in Table 2. The values of d 10 and s vary between 0.173-0.386 mm and 1.470-7.100 respectively.

Variation between friction factor and Reynolds number
The dimensionless flow parameters i.e., F r and R e were computed for soil samples using permeameter with different diameters i.e., 5.08, 10.16, & 15.24 cm. A logarithmic plot has been drawn between the dimensionless flow parameters as shown in Figure 3. The computation of F r and R e values plays a crucial role in determining the flow regime. The F r and R e are computed as: where, h i ¼ hydraulic gradient, V ¼ average flow velocity, d 50 ¼ mean grain-size, and υ ¼ fluid kinematic viscosity, & g ¼ gravitational constant. Figure 3 indicates straight-line variation between F r and R e 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., d 10 (Figure 4(a)), wherein a linear variation of K was observed with the different values of d 10 . From this investigation, it is concluded that as the value of d 10 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(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 s value. The observed value of s 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 & tends to zero, the curves approach asymptotically towards the y-axis as shown in Figure 4(b).

Hydraulic conductivity variation with σ/d 10
The variation of hydraulic conductivity computed from three permeameters was studied by plotting the K with different values of σ/d 10 . Figure 5 indicates that as the σ/d 10 value increases the K decreases. The curve touches asymptotically the lower limiting value and is concave upward. The inferences 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/d 10 values range between 3.80 and 41.06.
Investigations on the influence of σ/d 10 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 fifteen borehole soil samples. The grain-size parameters i.e., σ and d 10 have been used for model development. The developed statistical The standard form of the proposed K model is given below: where, l ¼ 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: 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 (z 0 , z 1 , z 2 , z 3 , & z 4 ) values are: The developed statistical model for the K estimation is:

Uncorrected Proof
Further, the σ/d 10 and K values computed experimentally and by using the developed statistical model have been given in Table 3.

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 & sorting coefficients, and porosity values were used to compute the K based on empirical equations. The value of grain-size parameters has been stated in Table 2. A specific value of the sorting coefficient has been used in each empirical equation except the Terzaghi equation. In Terzaghi's equation two different values of sorting coefficient i.e., 10.7 Â 10 À3 for smooth grains and 6.1 Â 10 À3 for coarse grains have been given, therefore an average sorting coefficient value of 8.4 Â 10 À3 has been used in the Terzaghi equation (Pucko & Verbovsek 2015). The computed values of hydraulic conductivity via empirical equation have been shown in Table 4.   indicated in Figure 6.  Figure 7 shows that the data points computed from the developed model are focused more on the agreement line as compared to Figure 6.
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.
The values of I a and R 2 vary from 0 to 1, and S i , 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 I a and R 2 , and lower values of S i , 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 S i , MAE, RMSE, and BIAS, and values are close to 1 for I a and R 2 for different diameter permeameters as shown in Table 5

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/d 10 ' for computing the K value. Therefore, the values of σ/d 10 for these soil samples have been determined as shown in Table 6.

Uncorrected Proof
By using the σ/d 10 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.

CONCLUSIONS
The present study is devoted to establish a statistical model for the computation of the hydraulic conductivity based on the grain-size parameters i.e., d 10 & σ. The developed model includes a factor 'λ' which incorporates the porous media