Assessment of Hydraulic Conductivity of Porous Media Using Empirical Relationships

Abstract

Flow-through porous media is concerned with the term hydraulic conductivity (K), which imparts a crucial role in the groundwater processes. The present work examines the impact of key parameters i.e., grain size and porosity on the K of four borehole soil samples (Gravelly, Coarse, Medium, and Fine sands) and evaluates the applicability of seven empirical relationships for K estimation. Experimental investigations postulate that an increase in the grain size and porosity value increases the K value. Further, the K values computed using the Kozeny–Carman relationship proved to be the best estimator for Coarse, medium, and fine sands followed by Beyer and Hazen relationships. However, the Beyer relationship had a closer agreement with experimentally obtained value for Gravelly sand. Alyamani and Sen relationship is very sensitive toward the grain-size curve pattern, hence it should be used carefully. Whereas other relationships considered in this study underestimated the K of all samples.

1. Introduction

Water is a vital natural resource, imperative for the existence of all living organisms. Various processes i.e., agricultural processes, groundwater management practices, and environmental quality are influenced by water [1]. Apart from agriculture, some other usages are municipal and industrial water supply, hydroelectric power, forestry, and navigation [2]. Water is available in different forms i.e., surface water, groundwater, ice caps, and glaciers, however, groundwater seems to be a more consistent source of water [3]. Investigation on the computation of hydraulic conductivity of borehole soil samples results in a potential alternative for groundwater monitoring [4].

Initially, Darcy’s law defines the term K as the 1-D flow of water through the saturated porous sediments [5]. K is the dominant hydraulic parameter of the porous media used for predicting the movement of fluid through the connecting voids [6]. It has a significant role, in estimating the quantity of seepage through earth dams and levees and conducting stability analysis of earth structures subjected to seepage forces [7]. Saturated K of porous media is important for modeling the flow of water in the saturated zone [8, 9]. In the previous studies, it was postulated that the K of granular porous media is related to grain size characteristics i.e., d₁₀, d₂₀, d₅₀, and d₆₀ [10]. This relationship is very convenient for hydraulic conductivity estimation in the initial stages of aquifer investigation. The representative grain size of porous media from the gradation analysis is helpful in the assessment of K values [11]. Various properties influenced the K of porous sediments i.e., porosity, structure alignment as well as different properties of fluid such as temperature and viscosity [12].

In groundwater investigations processes, there are many techniques namely laboratory and field methods, and empirical equations are available to estimate the K of porous sediments [13]. Precise knowledge of aquifer geometry and boundaries consequences the limited use of field methods [14]. Also, the collection of undisturbed soil samples is a challenging factor concerning the laboratory experimental techniques. Therefore, the computation of K values using empirical relationships has been used as a substitute to overcome the issues that occur due to the field and laboratory techniques [15, 16]. Various investigators derive the empirical relationships to compute the K value and should be used within particular domains of applicability [17]. The computed K values based on different empirical relationships to the similar size of porous sediments can result in different K values because the applicability and domains are different for individual empirical relationships [18].

Kasenow [19] analyzed some important empirical relationships on the same porous media and concluded that different K values may be obtained. Carrier [20] concluded that the Kozeny–Carman equation is the best estimator of K as compared to other empirical relationships. Odong [17] focused on the evaluation of K of porous media using empirical relationships and concluded that precise estimation of K is based on the Kozeny–Carman equation, however other relationships in the study overestimated the K values. Rosas et al. [6] estimated and compared K with empirical relationships for 400 samples of sediments with different grain size distributions. Cabalar & Akbulut [21] determined the hydraulic conductivity of sand samples of different shapes and grain sizes and evaluated them with empirical relationships. An M5 model tree was developed to predict K based on gradation analysis by Naeej et al. [22]. Ríha et al. [16] evaluated the applicability and reliability of glass beads of different diameters and assessed the K of glass beads using empirical relationships. Hong et al. [23] revised the Kozeny–Carman relationship based on effective void ratio and specific surface area and then used it to predict the hydraulic conductivity of the soil.

The literature revealed that different investigators use the existing empirical relationships to estimate the hydraulic conductivity values and vaguely define their applicability boundaries via normal description of materials used without suitable assessment and grain size distribution analysis. The present study has been focused to address this research gap. The main objectives of the study are:

1. To study the influence of key parameters namely grain size and porosity on the K of the borehole samples.

2. To determine the flow regime by analyzing the variations between friction factor and Reynolds number.

3. To evaluate the applicability of seven established empirical relationships for K estimation.

2. Materials and experimental procedure

In the present study, four representative soil samples were collected during borehole drilling operation at the Una district of Himachal Pradesh in India. Samples (1, 2, 3, and 4) were collected at an interval of 3 m. The collected samples were tested for gradation analysis as per standard procedure to determine different grain sizes i.e., d₁₀, d₂₀, d₅₀, and d₆₀ [24]. Samples 1, 2, 3, and 4 containing coarse porous media particles, hence subjected to the dry sieve analysis. A pycnometer test has been conducted on each collected soil sample to determine the specific gravity.

The K of soil samples was measured using a constant head permeameter having a diameter of 153 mm and a test length of 46.5 cm as shown in Figure 1. Initially, the samples were placed inside an oven at 105°C for about 24 hours for maturing and then added to the permeameter in a completely dry state and were compacted in layers with a rubber mallet. The upper part of the permeameter is connected to a water supply tank, which is situated at a height of 2.5 m above the permeameter, and the lower part is connected to an outlet pipe for discharge measurement. Before determining the K, the sample was saturated to maintain a steady flow condition. A constant head was preserved in the manometer pipe, and then the discharge value was measured for a fixed time interval. Five to six measurements were made at different constant heads. The average of the discharge values was taken to determine the K of the sample [25]. The water temperature was measured at the start and the end of the permeameter test. The value of K was calculated by multiplying the flow rate (cm³/s) by the specimen thickness (cm) and then diving it by the permeameter area (cm²) times the constant head (cm).

Figure 1.

Hydraulic conductivity measuring setup.

3. Established empirical relationships

From gradation analysis, effective grain diameter and particle uniformity are used to estimate the K using empirical relationships which relate the K value with the size property of porous media. Based on the previous investigations in the field of K computation, Vukovic and Soro [18] formulated a generalized K equation as:

$$K=\frac{g}{\vartheta }\ast \alpha \ast f\left(x\right)\ast d_x^2$$E1

where, g = gravitational constant, ϑ = Kinematic viscosity, α = sorting coefficient, x = porosity and, d_x = effective grain size. The values of $\alpha$, f(x), and d_x depend on different procedures used for gradation analysis. Vukovic and Soro [18] postulated a standard equation between uniformity coefficient (U) and porosity as:

$$x=0.255\left\{1+{0.83}^U\right\}$$E2

$$U=\frac{d_{60}}{d_{10}}$$E3

where, d₆₀ and d₁₀ are the grain diameter in (mm).

Several investigators developed different empirical relationships based on the standard equation as mentioned in Eq. (1). Table 1 represents the seven empirical relationships, which are used in the present work for the computation of K values of porous sediments.

Researcher	Relationship	α	Use
Hazen [26]	KHazen=$\frac{g}{v}.\alpha .\left[1+10\left(x-0.26\right)\right].d_{10}^2$	6 × 10^−4	0.1 mm < d10 < 3 mm; U < 5
Slichter [27]	KSlichter$=\frac{g}{v}.\alpha .x^{3.287}{d}_{10}^2$	1 × 10^−2	0.01 mm < d10 < 5 mm
Terzaghi [28]	KTerzaghi=$\frac{g}{v}.\alpha .{\left(\frac{x-0.13}{\sqrt[3]{1-x}}\right)}^2{d}_{10}^2$	8.4 × 10^−3	Large grain sand
Kozeny [29]-Carman [30, 31]	KK-C=$\frac{g}{v}.\alpha .\left[\frac{x^3}{{\left(1-x\right)}^2}\right].d_{10}^2$	8.3 × 10^−3	d10 < 3.0 mm suitable for gravel, sand, and silty soil
Beyer [32]	KBeyer=$\frac{g}{v}.\alpha .log\frac{500}{U}{d}_{10}^2$	6 × 10^−4	0.06 mm < d10 < 0.6 mm; 1 < U < 20
USBR [33]	KUSBR=$\frac{g}{v}.\alpha .d_{20}^{2.3}$	4.8 × 10^−4	U < 5; Medium grained sand
Alyamani and Sen [34]	KA/S$=\alpha {\left[I+0.025\left(d_{50}-d_{10}\right)\right]}^2$	1300	Well distributed sample

Table 1.

Empirical relationships for hydraulic conductivity estimation.

U = uniformity coefficient, d₁₀ and d₅₀ = grain size (mm) and, I = line intercept in mm formed by d₅₀ and d₁₀ with the grain size axis.

4. Results and discussion

Gradation analysis has been conducted on the collected soil samples to compute the gradation characteristics namely grain size, uniformity coefficient, and porosity. The influence of porosity and various grain sizes on the K of soil samples was investigated. Further, the experimentally measured K values were compared with the values determined via the empirical relationships.

5. Gradation analysis

Initially, the gradation analysis has been performed on the collected soil samples using a mechanical sieve device. Figure 2 shows the gradation curve for different borehole soil samples. The gradation curve analysis helps to categorize the soil samples based on particle size as shown in Table 2 [35].

Figure 2.

Gradation curve of collected samples.

From the gradation curve, grain size at 10%, 20%, 50%, and 60% cumulative weight was determined. The uniformity coefficient, intercept, and porosity values for samples 1, 2, 3, and 4 are given in Table 3.

5.1 Variation of hydraulic conductivity with grain size and porosity

From the gradation analysis, the grain size i.e., d₁₀, d₂₀, d_50, and d₆₀ for all samples were determined. A linear variation between hydraulic conductivity and effective grain size (d₁₀) was observed. From this, it is concluded that as the value of effective grain size increases the hydraulic conductivity also increases. The variation falls on the same lines for other gain sizes i.e., (d₂₀, d₅₀, and d₆₀) as shown in Figure 3. The porosity (x) for all samples was determined and plotted against hydraulic conductivity as shown in Figure 4. A straight line obtained between these two parameters indicates that K increases as the porosity value increases, which is in line with the findings of Fallico [36].

Figure 3.

Variation of hydraulic conductivity with grain size.

Figure 4.

Variation of hydraulic conductivity with porosity (x).

5.2 Flow regime analysis

To govern the flow regime the variation between dimensionless quantities i.e., friction factor (Fr) and Reynolds number (Re) were studied and plotted on a logarithmic scale. The standard equation to compute the Fr and Re are given as:

$$\mathrm{Fr}=\frac{h_i\ast g\ast d_{50}\ast 2}{U^2}$$E4

$$Re=\frac{U{d}_{50}}{\phi }$$E5

where, h_i = hydraulic gradient, d₅₀ = average size, g = gravitational constant, U = flow velocity, and $\phi$ = fluid kinematic viscosity.

From Figure 5, a linear plot between Fr and Re was observed having the Reynolds number value less than 1 [37], which validates the flow regime to be Darcy’s or linear regime.

Figure 5.

Plot between Fr and Re of collected soil samples.

5.3 Computation of K using empirical relationships

Initially, the K of borehole samples was determined using a constant head permeameter. For Gravelly, Coarse, Medium and Fine sands the obtained K value was found to be 0.152, 0.128, 0.072, and 0.052 cm/s respectively. From gradation analysis different parameters i.e., grain size, uniformity coefficient, intercept, and porosity values were determined, which has been used to compute the K of all samples using seven empirical relationships. The value of kinematic viscosity i.e., 0.885 mm²/s derived at a temperature of 27°C was used in the estimation of K using empirical relationships. The computed K value of all samples using empirical relationships is given in Table 4.

Sample	Composition	Classification
1	6% medium gravel, 20% fine gravel, 35% coarse sand, 25% medium sand, and 14% fine sand	Gravelly sand
2	12% fine gravel, 40% coarse sand, 43% medium sand, and 5% fine sand	Coarse sand
3	2% coarse sand, 84% medium sand, and 14% fine sand	Medium sand
4	48% medium sand and 52% fine sand	Fine sand

Table 2.

Soil samples classification based on gradation curve.

Sample – its classification	d10 (mm)	d20 (mm)	d50 (mm)	d60 (mm)	I (intercept) (mm)	(U) uniformity coefficient	(x) porosity
1 – Gravelly sand	0.330	0.465	1.160	1.750	0.255	5.30	0.425
2 – Coarse sand	0.316	0.404	0.950	1.380	0.249	4.36	0.412
3 – Medium sand	0.180	0.230	0.390	0.480	0.154	2.67	0.390
4 – Fine sand	0.165	0.190	0.260	0.386	0.135	2.34	0.382

Table 3.

Grain size and other important properties of samples.

Sample – its classification	Kexp (cm/s)	KHazen (cm/s)	KSlichter (cm/s)	KTerzaghi (cm/s)	KK-C (cm/s)	KBeyer (cm/s)	KUSBR (cm/s)	KA/S (cm/s)
1 – Gravelly sand	0.152	NA	0.072	0.093	0.233	0.147	NA	0.114
2 – Coarse sand	0.128	0.167	0.059	0.076	0.132	0.136	NA	0.105
3 – Medium sand	0.072	0.050	0.016	NA	0.064	0.049	0.018	0.038
4 – Fine sand	0.052	0.040	0.013	NA	0.046	0.041	0.012	0.028

Table 4.

Estimation of hydraulic conductivity using empirical relationships.

NA = not applicable.

Hazen and USBR empirical relationships are irrelevant to estimate the K value of gravelly sand because the value of uniformity coefficient is greater than 5. For medium and fine sands, the Terzaghi relationship was not used because it is relevant only for large grain sand [14]. Also, the USBR equation applies only to the sizes of medium sand and is thus irrelevant for coarse sand.

Slichter, Terzaghi, and USBR relationships underestimate the K values, which is consistent with the findings of Cheng and Chen [38]. Alyamani and Sen relationship results in relatively good prediction for Gravelly and Coarse sands, but it underestimates the K for medium and fine sands, because of their poor grading. The K values computed using the Kozeny–Carman relationship for coarse, medium, and fine sands have a closer agreement with the measured values followed by Beyer and Hazen relationships as shown in Figure 6. The K-C relationship underestimated the K value for Gravelly sand because the relationship is not suitable if the grain size distribution has a flat, long tail of fine fraction [20]. Beyer relationship provides better K prediction for gravely sand.

Figure 6.

Comparison of measured and empirically computed hydraulic conductivity.

6. Conclusions

The present work is focused to evaluate seven established empirical relationships for estimating the K of borehole soil samples. Hydraulic conductivity estimation using gradation analysis can also lead to underestimation until the relevant empirical relationship is used. The study examines the impact of key parameters i.e., grain size and porosity on the K value. From the experimental study, it has been observed that with the increase in the grain size and porosity, the K of borehole soil samples increases. The computed K values using the Kozeny–Carman relationship have a closer agreement with the measured values for coarse, medium, and fine sands followed by Beyer and Hazen relationships. Notably, the Beyer relationship provides a better K prediction for Gravelly sand. Alyamani and Sen relationship depends on the grain size curve pattern and should be used carefully. Other relationships i.e., Slichter, Terzaghi, and USBR manifestly underestimate the K of borehole samples.