Grain-size model for hydraulic conductivity estimation of porous media

The hydraulic conductivity (k) of porous media and its accurate assessment is important for a hydrogeologist in aquifer and groundwater studies (Leroueil et al. 2002). Initially, Henry Darcy explained the concept of hydraulic conductivity and defined it as the ease with which the flow of fluid takes place through the interconnected pore space (Pucko & Verbovsek 2015; Chandel et al. 2022a). For porous media, the studies concerning the k estimation are significant to govern the percolation ability of the permeable bed and soil sediments, as these factors are reliant on groundwater recharge (Chandel et al. 2022b).

In the previous studies, various researchers namely Parvazinia et al. (2006), Kundu et al. (2016), and Singh et al. (2021) studied different approaches for modelling the k of flow through porous media. The outcomes of these studies infer some restrictions, i.e., simulating the infinite extent of sub-surface flow. One of the important considerations is to control other factors that influence the k of porous media while examining the influence of individual parameters, i.e., standard deviation and mean and effective size (Lu et al. 2012). Pliakas & Petalas (2011) developed a statistical regression model by examining the correlation between the k and grain-size parameters and postulated that the developed model performed better in the estimation of k values. Salarashayeri & Siosemarde (2012) proposed a hydraulic conductivity equation using grain diameters, i.e., d₁₀, d₅₀, and d₆₀. The study recommended that the grain diameter (d₁₀) is an influencing factor for the k estimation of porous media. The M5 model tree regression approach was used to develop a k model based on the grain-size data (Naeej et al. 2017). Wang et al. (2017) proposed a k model by performing the dimensional analysis between the particle size parameters with the k. The developed equation performs better in the estimation of k values as compared to the existing empirical models. Ren & Santamarina (2018) estimated the k of porous media using the sediment and index characteristics i.e, void ratio, grain-size distribution, and surface area. Toumpanou et al. (2021) evaluated the k of the crushed sand-sized limestone using the pre-existing six empirical equations. Mujtaba et al. (2021) correlated the hydraulic conductivity with the different grain-sizes of sandy soil and concluded that the d₁₀ grain-size shows better goodness of fit with the k as compared to the other grain-sizes. Khaja et al. (2022) developed an equation for the estimation of k of sandy aquifers using the grain-size data. The developed equation gives better results when the grain-size, i.e., d₃₀ is considered to be an effective parameter.

From the existing literature, it has been seen that limited research work has been done to examine the influence of d₁₀ and σ on the k of porous media. The standard deviation signifies the non-uniformity of the soil particles, whereas the effective grain-size represents the behavior of the entire grain-size pattern. The study incorporates two parameters, i.e., effective grain-size (d₁₀) and standard deviation (σ) for the development of a grain-size model. The present study is aimed:

• (1)

  To investigate the influence of d₁₀ and σ on the k value of porous media.

• (2)

  To develop a grain-size model using parameters, i.e., σ & d₁₀ and evaluate its efficacy in estimating the k of porous media.

• (3)

  To validate the grain-size model using data set of independent soil samples.

The study uses 27 borehole soil samples, which were attained during the ongoing drilling process from the Chamba district of Himachal Pradesh in India. Thin-walled sampler tubes were used to collect the undisturbed soil samples. For experimental work, the soil samples were collected at an interval of 3.5 m from the core material.

Figure 1 indicates the hydraulic conductivity measuring setup, which includes galvanized iron permeameters of different diameters. For each permeameter, the total and test lengths are 1 m and 0.46 m respectively. An overhead tank is provided above the permeameter outlet, which provides a continuous flow of water to the permeameter. The arrangement of pressure taping points is provided along the permeameter periphery, which helps in measuring the readings of the manometer. The volume of the water is measured using a measuring cylinder for a particular time interval. The standard methodology as described by Chandel & Shankar (2021) has been followed to determine the k of soil samples. During the k measurement, the temperature of the water is recorded at the start and end of the analysis using a digital thermometer (ASTM 2006).

Based on the literature review, various empirical models that are used to evaluate the k of porous media have been used in this study. These empirical models relate the k with the grain-size, viscosity of fluid, porosity, sorting coefficient, and uniformity coefficient. Table 1 shows the empirical models with their boundary conditions used in the present study to evaluate the k values. In the empirical models, the ν and g values were taken as 0.885 mm²/s and 981 cm/s² respectively.

Table 1

Empirical models for k assessment

Investigator .	Equation .	Boundary conditions .
Hazen (1892)		d10 (0.1–3mm) U<5
Kozeny (1927), Carman (1937, 1956)		d10 < 3.0 mm
Chapuis et al. (2005)		d10 (0.03–3 mm)
Naeej et al. (2017)		suitable for sand < 3.0 mm

Investigator .	Equation .	Boundary conditions .
Hazen (1892)		d10 (0.1–3mm) U<5
Kozeny (1927), Carman (1937, 1956)		d10 < 3.0 mm
Chapuis et al. (2005)		d10 (0.03–3 mm)
Naeej et al. (2017)		suitable for sand < 3.0 mm

where, U, uniformity coefficient i.e., d₆₀/d₁₀, and n, porosity.

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The hydraulic test and gradation analysis have been conducted on the 27 borehole soil samples. The data points of 15 soil samples were used to develop a k model by examining the influence of σ/d₁₀ parameter on the k. The data set of remaining independent soil samples was used for the validation of the developed model.

The grain-size analysis was performed to plot the grain-size curve between the particle size and percent finer. Figure 2 represents the grain-size curve for 15 soil samples. The grain sizes corresponding to the 10%, 30%, and 60% finer by weight i.e., d₁₀, d₃₀, and d₆₀, standard deviation, and uniformity coefficient values were determined using the grain-size curve. Whereas, the grain-size parameters (d₁₀ and σ) for the remaining soil samples have been determined and provided in the validation section.

Table 2 represents the basic characteristics of the 15 soil samples. The and d₁₀ values vary between 1.470 to 7.100 and 0.173 to 0.386 mm respectively.

Figure 3(b) indicates that the hydraulic conductivity decreases with the increase in the σ values, which vary from 1.47 to 7.10. The value more than unity signifies the non-uniformity of soil particles. In this study, the non-uniformity of soil particles increases with the increase in the σ value which results in providing more compactness to the porous particles and thus results in the decreased k value.

The hydraulic conductivity variation was examined by plotting the k with different σ/d₁₀ values for different diameter permeameters.

The data points of 15 soil samples have been used for the development of a grain-size model. For model development, the grain-size parameters (σ and d₁₀) have been integrated to form a dimensionless parameter ‘σ/d₁₀’. The equation of the developed model is derived by investigating the variations of the σ/d₁₀ parameter with the hydraulic conductivity values obtained from the three different diameter permeameters for 15 soil samples i.e., a total number of 45 k versus σ/d₁₀ variations were examined for model development. The ‘σ/d₁₀’ parameter in the developed model includes coefficients of degree 0–4. For the development of the model, the principle of least-squares approach is used in this study (Wang et al. 2017).

The observed β suggests that its β value is smaller for maximum and larger for minimum, permeameter size. The outcome of the study infers that the increase in the porous media extent results in an insignificant β value (Zieba 2017).

The values of empirical constants i.e., a₀, a₁, a₂, z₃, & a₄ are:

a₀ = 0.388, a₁ = 0.0327, a₂ = 0.0013, a₃ = 2.56 × 10⁻⁵, & a₄ = 2 × 10⁻⁷

Table 3 represents the developed model-based and experimentally measured k values for 15 soil samples.

Further, the uncertainty analysis of the developed model, i.e., Equation (9) has been performed (Parsaie & Haghiabi 2021). In Equation (9), the dependent parameter is k, whereas the independent parameters are σ & d₁₀. Initially, for uncertainty analysis, the σ value remains constant, while the d₁₀ value increases from the baseline value up to 20%, resulting in an increase in the k value by 5.5%. The k value decreases up to 6.2%, when the σ value increases from 0 to 20%, while the d₁₀ value remains constant. Based on the results, both the σ and d₁₀ parameters are dependent on k, while the d₁₀ is positively sensitive to k and σ is inversely sensitive to k.

In this study, four empirical models have been used to evaluate the k values. Table 2 represents the grain-size parameters that were used to calculate the k values via empirical models.

Further, various statistical indicators were used to evaluate the quantitative assessment of the developed model for different diameter permeameters as given in Table 4.

The BIAS, MAE, S_i, and RMSE values vary from 0 to ∞ and R² and I_a from 0 to 1 (Rosas et al. 2014). Lower values of MAE, S_i, BIAS, and RMSE, and values closer to 1 for I_a and R² point to a better agreement between the observed and measured parameters (Naeej et al. 2017). The BIAS, S_i, MAE, RMSE, R², and I_a values for the developed model are 0.006, 0.039, 0.005, 0.009, 0.978, and 0.976 for 5.08 cm, 0.003, 0.051, 0.008, 0.005, 0.984, and 0.974 for 10.16 cm, and 0.004, 0.041, 0.007, 0.006, 0.962, and 0.981 for 15.24 cm diameter permeameters respectively. The values of statistical indicators for the developed model establish its efficacy in estimating the k of porous media.

The grain-size parameters (d₁₀ and σ) of the remaining 12 soil samples have been determined and used for validation. For computing the k value, the developed model includes a parameter, i.e., σ/d₁₀. Therefore, for these 12 soil samples, the σ/d₁₀ values have been determined as shown in Table 5.

Figure 7 indicates a fairly good agreement between the measured and developed model-based predicted k values. For the developed model the BIAS, I_a, R², S_i, RMSE, and MAE values during validation are 0.038, 0.885, 0.934, 0.048, 0.016, and 0.028 respectively, which substantiate the performance of the grain-size model in estimating the hydraulic conductivity of porous media.

The present study is focused on developing a hydraulic conductivity model using grain-size parameters i.e., d₁₀ and σ. The grain-size model comprises a factor ‘β’ which inherits the particle roughness, porous media extent, and compactness of porous media. The increase in the porous media extent results in an insignificant β value. For borehole soil samples, the influence of d₁₀ and σ on the hydraulic conductivity elucidate that the k increases with the increase in the d₁₀ grain-size and decreases with the increase in the σ value. The evaluation of k values via empirical models infers that the Hazen model performs well in k estimation, followed by the Kozeny-Carman model. The quantitative assessment of the grain-size model via the statistical indicators substantiates its efficacy in estimating the k. Also, using the independent data set, the study validates the performance of the grain-size model in estimating k. The study recommends further investigations to examine the influence of particle and wall roughness on the k of porous media.

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

The authors declare there is no conflict.