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Assessment of Hydraulic Conductivity of Porous Media Using Empirical Relationships

Written By

Abhishish Chandel and Vijay Shankar

Submitted: January 25th, 2022 Reviewed: February 8th, 2022 Published: March 25th, 2022

DOI: 10.5772/intechopen.103127

Sediment Transport Edited by Davide Pasquali

From the Edited Volume

Sediment Transport [Working Title]

Dr. Davide Pasquali

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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.


  • empirical relationship
  • grain size
  • hydraulic conductivity
  • porosity

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 Kas the 1-D flow of water through the saturated porous sediments [5]. Kis 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 Kof 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 Kof 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 Kvalues [11]. Various properties influenced the Kof 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 Kof 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 Kvalues 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 Kvalue and should be used within particular domains of applicability [17]. The computed Kvalues based on different empirical relationships to the similar size of porous sediments can result in different Kvalues 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 Kvalues may be obtained. Carrier [20] concluded that the Kozeny–Carman equation is the best estimator of Kas compared to other empirical relationships. Odong [17] focused on the evaluation of Kof porous media using empirical relationships and concluded that precise estimation of Kis based on the Kozeny–Carman equation, however other relationships in the study overestimated the Kvalues. Rosas et al. [6] estimated and compared Kwith 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 Kbased 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 Kof 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 Kof 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 Kestimation.


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 Kof 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 Kof the sample [25]. The water temperature was measured at the start and the end of the permeameter test. The value of Kwas 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 Kusing empirical relationships which relate the Kvalue with the size property of porous media. Based on the previous investigations in the field of Kcomputation, Vukovic and Soro [18] formulated a generalized Kequation as:


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


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 Kvalues of porous sediments.

Hazen [26]KHazen=gv.α.1+10x0.26.d1026 × 10−40.1 mm < d10 < 3 mm
U < 5
Slichter [27]KSlichter=gv.α.x3.287d1021 × 10−20.01 mm < d10 < 5 mm
Terzaghi [28]KTerzaghi=gv.α.x0.131x32d1028.4 × 10−3Large grain sand
Kozeny [29]-Carman [30, 31]KK-C=gv.α.x31x2.d1028.3 × 10−3d10 < 3.0 mm suitable for gravel, sand, and silty soil
Beyer [32]KBeyer=gv.α.log500Ud1026 × 10−40.06 mm < d10 < 0.6 mm
1 < U < 20
USBR [33]KUSBR=gv.α.d202.34.8 × 10−4U < 5
Medium grained sand
Alyamani and Sen [34]KA/S=αI+0.025d50d1021300Well 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 Kof soil samples was investigated. Further, the experimentally measured Kvalues 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, d50, 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 Kincreases 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:


where, hi = hydraulic gradient, d50 = average size, g = gravitational constant, U = flow velocity, and φ= 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 Kusing empirical relationships

Initially, the Kof borehole samples was determined using a constant head permeameter. For Gravelly, Coarse, Medium and Fine sands the obtained Kvalue 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 Kof 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 Kusing empirical relationships. The computed Kvalue of all samples using empirical relationships is given in Table 4.

16% medium gravel, 20% fine gravel, 35% coarse sand, 25% medium sand, and 14% fine sandGravelly sand
212% fine gravel, 40% coarse sand, 43% medium sand, and 5% fine sandCoarse sand
32% coarse sand, 84% medium sand, and 14% fine sandMedium sand
448% medium sand and 52% fine sandFine sand

Table 2.

Soil samples classification based on gradation curve.

Sample – its classificationd10 (mm)d20 (mm)d50 (mm)d60 (mm)I(intercept) (mm)(U) uniformity coefficient(x) porosity
1 – Gravelly sand0.3300.4651.1601.7500.2555.300.425
2 – Coarse sand0.3160.4040.9501.3800.2494.360.412
3 – Medium sand0.1800.2300.3900.4800.1542.670.390
4 – Fine sand0.1650.1900.2600.3860.1352.340.382

Table 3.

Grain size and other important properties of samples.

Sample – its classificationKexp (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 sand0.152NA0.0720.0930.2330.147NA0.114
2 – Coarse sand0.1280.1670.0590.0760.1320.136NA0.105
3 – Medium sand0.0720.0500.016NA0.0640.0490.0180.038
4 – Fine sand0.0520.0400.013NA0.0460.0410.0120.028

Table 4.

Estimation of hydraulic conductivity using empirical relationships.

NA = not applicable.

Hazen and USBR empirical relationships are irrelevant to estimate the Kvalue 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 Kvalues, 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 Kfor medium and fine sands, because of their poor grading. The Kvalues 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 Kvalue 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 Kprediction 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 Kof 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 Kvalue. From the experimental study, it has been observed that with the increase in the grain size and porosity, the Kof borehole soil samples increases. The computed Kvalues 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 Kprediction 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 Kof borehole samples.


  1. 1. Kiran DA, Ramaraju HK. Assessment of water and soil quality along the coastal region of Mangaluru. Journal of Indian Association for Environmental Management (JIAEM). 2019;39(1–4):9-13
  2. 2. Loucks DP, Van Beek E. Water Resource Systems Planning and Management: An Introduction to Methods, Models, and Applications. Springer. Paris: UNESCO; 2017
  3. 3. Alabi AA, Bello R, Ogungbe AS, Oyerinde HO. Determination of groundwater potential in Lagos State University, Ojo; using geoelectric methods (vertical electrical sounding and horizontal profiling). Report and Opinion. 2010;2(5):68-75
  4. 4. Anomohanran O. Geophysical investigation of groundwater potential in Ukelegbe, Nigeria. Journal of Applied Sciences. 2013;13(1):119-125
  5. 5. Deb SK, Shukla MK. Variability of hydraulic conductivity due to multiple factors. American Journal of Environmental Sciences. 2012;8(5):489
  6. 6. Rosas J, Lopez O, Missimer TM, Coulibaly KM, Dehwah AH, Sesler K, et al. Determination of hydraulic conductivity from grain-size distribution for different depositional environments. Groundwater. 2014;52(3):399-413
  7. 7. Ríha J. Groundwater flow problems and their modelling. In: Assessment and Protection of Water Resources in the Czech Republic. Cham: Springer; 2020. pp. 175-199
  8. 8. Chandel A, Shankar V, Alam MA. Experimental investigations for assessing the influence of fly ash on the flow through porous media in Darcy regime. Water Science and Technology. 2021;83(5):1028-1038
  9. 9. Perkins KS, Elango L. Measurement and modeling of unsaturated hydraulic conductivity. In: Hydraulic Conductivity–Issues, Determination and Applications. China: Intech; 2011. pp. 419-434
  10. 10. Sperry JM, Peirce JJ. A model for estimating the hydraulic conductivity of granular material based on grain shape, grain size, and porosity. Groundwater. 1995;33(6):892-898
  11. 11. Boadu FK. Hydraulic conductivity of soils from grain-size distribution: New models. Journal of Geotechnical and Geoenvironmental Engineering. 2000;126(8):739-746
  12. 12. Omojola AD, Akinpelu SJ, Adesegun AM, Akinyemi OD. A micro study to determine porosity, hydraulic conductivity, permeability and the discharge rate of groundwater in Ondo state riverbeds, southwestern Nigeria. International Journal of Geosciences. 2014;5(11):1254
  13. 13. Todd DK, Mays LW. Groundwater Hydrology. New York: John Wiley & Sons; 2005
  14. 14. Ishaku JM, Gadzama EW, Kaigama U. Evaluation of empirical formulae for the determination of hydraulic conductivity based on grain-size analysis. Journal of Geology and Mining Research. 2011;3(4):105-113
  15. 15. Cirpka OA. Environmental fluid mechanics I: Flow in natural hydrosystems. Journal of Hydrology. 2003;283:53-66
  16. 16. Ríha J, Petrula L, Hala M, Alhasan Z. Assessment of empirical formulae for determining the hydraulic conductivity of glass beads. Journal of Hydrology and Hydromechanics. 2018;66(3):337-347
  17. 17. Odong J. Evaluation of empirical formulae for determination of hydraulic conductivity based on grain-size analysis. Journal of American Science. 2007;3(3):54-60
  18. 18. Vukovic M, Soro A. Determination of Hydraulic Conductivity of Porous Media from Grain-Size Composition. Littleton, Colorado: Water Resources Publications; 1992 [551.49 V 986]
  19. 19. Kasenow M. Determination of Hydraulic Conductivity from Grain Size Analysis. LLC, Highland Ranch, CO, USA: Water Resources Publication; 2002. p. 83
  20. 20. Carrier WD. Goodbye, hazen; hello, kozeny-carman. Journal of Geotechnical and Geoenvironmental Engineering. 2003;129(11):1054-1056
  21. 21. Cabalar AF, Akbulut N. Evaluation of actual and estimated hydraulic conductivity of sands with different gradation and shape. Springerplus. 2016;5(1):820
  22. 22. Naeej M, Naeej MR, Salehi J, Rahimi R. Hydraulic conductivity prediction based on grain-size distribution using M5 model tree. Geomechanics and Geoengineering. 2017;12(2):107-114
  23. 23. Hong B, Li XA, Wang L, Li L, Xue Q, Meng J. Using the effective void ratio and specific surface area in the Kozeny–Carman equation to predict the hydraulic conductivity of loess. Water. 2020;12(1):24
  24. 24. ASTM. Standard D422—Particle-Size Analysis of Soils. PA, USA: West Conshohocken; 2007
  25. 25. ASTM. Standard D2434—Permeability of Granular Soils (Constant Head). PA, USA: West Conshohocken; 2006
  26. 26. Hazen A. Some physical properties of sands and gravels, with special reference to their use in filtration. In: 24th Annual Rep., Massachusetts State Board of Health, Pub. Doc. No. 34. 1892. pp. 539-556
  27. 27. Slichter CS. Theoretical investigation of the motion of ground waters. In: The 19th Ann. Rep. US Geophys Survey. 1899. pp. 304-319
  28. 28. Terzaghi KARL. Principles of soil mechanics. Engineering News-Record. 1925;95(19–27):19-32
  29. 29. Kozeny J. Uber kapillare leitung der wasser in Boden. Royal Academy of Science, Vienna, Proceedings, Class I. 1927;136:271-306
  30. 30. Carman PC. Flow of Gases Through Porous Media. London: Butterworths Scientific Publications; 1956
  31. 31. Carman PC. Fluid flow through granular beds. Transactions. Institute of Chemical Engineers. 1937;15:150-166
  32. 32. Beyer W. On the determination of hydraulic conductivity of gravels and sands from grain-size distributions. Wasserwirtschaft Wassertechnik. 1964;14(6):165-169
  33. 33. Mallet C, Pacquant J. Les barrages en terre. Paris: Editions Eyrolles; 1951. p. 345
  34. 34. Alyamani MS, Şen Z. Determination of hydraulic conductivity from complete grain-size distribution curves. Groundwater. 1993;31(4):551-555
  35. 35. ASTM. Standard practice for classification of soils for engineering purposes (unified soil classification system). In: Annual Book of ASTM Standards. West Conshohocken, PA: ASTM, International; 2010
  36. 36. Fallico C. Reconsideration at field scale of the relationship between hydraulic conductivity and porosity: The case of a sandy aquifer in South Italy. The Scientific World Journal. 2014;2014:1-15
  37. 37. Chandel A, Shankar V. Evaluation of empirical relationships to estimate the hydraulic conductivity of borehole soil samples. ISH Journal of Hydraulic Engineering. 2021:1-10
  38. 38. Cheng C, Chen X. Evaluation of methods for determination of hydraulic properties in an aquifer–aquitard system hydrologically connected to a river. Hydrogeology Journal. 2007;15(4):669-678

Written By

Abhishish Chandel and Vijay Shankar

Submitted: January 25th, 2022 Reviewed: February 8th, 2022 Published: March 25th, 2022