Open Access

Insights Into Estimation of Sand Permeability: From Empirical Relations to Microstructure-based Methods

Bartłomiej Bodak
Bartłomiej Bodak
 and 
Maciej Sobótka
Maciej Sobótka
   | Mar 29, 2024
Studia Geotechnica et Mechanica's Cover Image
About this article
Previous Article
Next Article
Cite
Article Category: Original Study
Published Online: Mar 29, 2024
Page range: 1 - 20
DOI: https://doi.org/10.2478/sgem-2024-0001
Keywords
© 2024 Bartłomiej Bodak et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1:

Grain size distribution curves of analyzed samples.
Grain size distribution curves of analyzed samples.

Figure 2:

Setup for measurement.
Setup for measurement.

Figure 3:

Permeameter fixture.
Permeameter fixture.

Figure 4:

General concept of a pore-network model.
General concept of a pore-network model.

Figure 5:

Analogous model of the resistor network.
Analogous model of the resistor network.

Figure 6:

Rendered view of reconstructed a) sample 1, b) sample 2, and c) sample 3.
Rendered view of reconstructed a) sample 1, b) sample 2, and c) sample 3.

Figure 7:

Exemplary slice, volumes of interest and binarized image of a) sample 1, b) sample 2, and c) sample 3.
Exemplary slice, volumes of interest and binarized image of a) sample 1, b) sample 2, and c) sample 3.

Figure 8:

Results of measurements in permeameter and best-fitting theoretical curves: a) sample 1, b) sample 2, and c) sample 3, and d) reference run without the sample attached. The vertical axis is scaled logarithmically for better fitting evaluation.
Results of measurements in permeameter and best-fitting theoretical curves: a) sample 1, b) sample 2, and c) sample 3, and d) reference run without the sample attached. The vertical axis is scaled logarithmically for better fitting evaluation.

Figure 9:

Comparison of measured and simulated grain size distribution curves from different sizes of VOI for a) sample 1, b) sample 2, and c) sample 3.
Comparison of measured and simulated grain size distribution curves from different sizes of VOI for a) sample 1, b) sample 2, and c) sample 3.

Figure 10:

Relative differences between hydraulic conductivity calculated with data from simulated sifting and those from granulometric analysis.
Relative differences between hydraulic conductivity calculated with data from simulated sifting and those from granulometric analysis.

Figure 11:

Tracks of random walkers after 1250 time steps in sample 3. Only 10% of all workers are shown for clarity.
Tracks of random walkers after 1250 time steps in sample 3. Only 10% of all workers are shown for clarity.

Figure 12:

Pore network extracted from a) sample 1, b) sample 2, and c) sample 3 with a zoomed fragment of the network.
Pore network extracted from a) sample 1, b) sample 2, and c) sample 3 with a zoomed fragment of the network.

Figure 13:

Streamlines of flow calculated using LBM: a) sample 1, b) sample 2, and c) sample 3.
Streamlines of flow calculated using LBM: a) sample 1, b) sample 2, and c) sample 3.

Figures 14:

Calculated and measured hydraulic conductivities for a) sample 1, b) sample 2, and c) sample 3.
Calculated and measured hydraulic conductivities for a) sample 1, b) sample 2, and c) sample 3.

Results of simulations using the lattice-Boltzmann method.

Sample no. Sample name VOI size Porosity derived from image data Permeability Hydraulic conductivity at 10°C
[vx] φimg [−] k [μm2] K [m/s]
1 Fine sand 4003 0.365 23.489 1.758E-4
6003 0.364 17.567 1.317E-4
2 Fine sand with lignite 4003 0.511 20.923 1.565E-4
6003 0.511 23.193 1.736E-4
3 Medium sand 4003 0.309 16.778 1.259E-4
6003 0.317 15.396 1.151E-4

Results of measurements with the described small-scale permeameter setup.

Sample no. Sample name Mean conductivity derived from the best-fit curve Conductivity of the apparatus Hydraulic conductivity in the measurement temperature Hydraulic conductivity at 10°C
Kequiv [m/s] Kap [m/s] Kex [m/s] Kcorr [m/s]
1 Fine sand 2.663E-5 4.927E-3 2.678E-5 1.951E-5
2 Fine sand with lignite 4.457E-6 4.461E-6 3.250E-6
3 Medium sand 6.183E-5 6.262E-5 4.562E-5

Results of simulations using the pore-network modeling approach.

Sample no. Sample name VOI size Porosity derived from image data Permeability Hydraulic conductivity at 10°C
[vx] φimg [−] k [μm2] K [m/s]
1 Fine sand 4003 0.365 23.666 1.786E-4
6003 0.364 23.587 1.780E-4
8003 0.363 24.061 1.816E-4
2 Fine sand with lignite 4003 0.511 28.433 2.145E-4
6003 0.511 27.969 2.110E-4
8003 0.506 27.338 2.063E-4
3 Medium sand 4003 0.309 17.311 1.306E-4
6003 0.317 20.301 1.532E-4
8003 0.317 22.087 1.667E-4

Measured properties of the samples.

Sample no. Sample name Soil type according to PN-EN ISO 14688-2:2018 Bulk density Specific density Porosity in loose state Hydraulic conductivity in falling-head test at 10°C Uniformity coefficient U=d60/d10 GSD curve slope coefficient C=d302/(d60·d10)
[−] ρ [g/cm3] ρs [g/cm3] φ [−] K [m/s] U [−] C [−]
1 Fine sand FSa 1.549 2.634 0.412 1.702E-5 1.840 1.054
2 Fine sand with lignite FSa 1.238 2.644 0.532 3.189E-6 2.532 1.027
3 Medium sand MSa 1.652 2.654 0.377 4.067E-5 3.147 1.003

Results of estimation using the Kozeny–Carman equation.

Sample no. Sample name VOI size Porosity derived from image data Tortuosity in direction of the flow Specific surface area per unit volume Permeability Hydraulic conductivity at 10°C
[vx] φimg [−] τ [−] S [1/m] k [μm2] K [m/s]
1 Fine sand 4003 0.365 1.937 38748 16.587 1.242E-4
6003 0.364 1.935 38447 16.674 1.249E-4
8003 0.363 1.897 37820 17.378 1.301E-4
2 Fine sand with lignite 4003 0.511 1.732 72321 24.639 1.845E-4
6003 0.511 1.722 73061 24.283 1.818E-4
8003 0.506 1.763 72932 22.645 1.696E-4
3 Medium sand 4003 0.309 2.009 40988 7.323 5.484E-4
6003 0.317 1.980 40043 8.604 6.443E-4
8003 0.317 1.946 37079 10.209 7.645E-4

Summary of used empirical formulae.

Method Equation form Coefficient C or C′ Porosity function f(φ) Effective diameter de Exponent m Applicability
Seelheim (1880) (5) 3570 1 d50 2 Sands and clays
Hazen (1911) (4) 6.0E-4 1+10(φ−0.26) d10 2 0.1 mm<d10<3 mm*
U<5
Sauerbrey (1932) (4) 3.75E-3 φ3/(1−φ)2 d17 2 d17<5 mm
USBR (Říha et al., 2018) (4) 4.8E-4·(1000d20)0.3 1 d20 2 U<5
Beyer (1964) (4) 6E-4·log(500/U) 1 d10 2 0.06 mm<d10<0.6 mm
1<U<20
Chapuis et al. (2005) (5) 1219.9 φ2.3475/(1−φ)1.565 d10 1.565 0.03 mm<d10<3 mm

Results of estimation using empirical equations.

Sample no. Sample name Method Effective diameter Effective diameter value Hydraulic conductivity at 10°C
[−] [−] de [mm] K [m/s]
1 Fine sand Seelheim d50 0.273 2.661E-4
Hazen d10 0.163 3.031E-4
Sauerbrey d17 0.189 2.045E-4
USBR d20 0.201 8.937E-5
Beyer d10 0.163 2.927E-4
Chapuis d10 0.163 4.125E-4
2 Fine sand with lignite Seelheim d50 0.138 6.799E-5
Hazen d10 0.062 N/A
Sauerbrey d17 0.077 1.151E-4
USBR d20 0.082 1.150E-5
Beyer d10 0.062 3.994E-5
Chapuis d10 0.062 2.363E-4
3 Medium sand Seelheim d50 0.381 5.182E-4
Hazen d10 0.143 2.01E-4
Sauerbrey d17 0.179 1.254E-4
USBR d20 0.196 8.531E-5
Beyer d10 0.143 2.037E-4
Chapuis d10 0.143 2.496E-4
eISSN:
2083-831X
Language:
English
Publication timeframe:
4 times per year
Journal Subjects:
Geosciences, other, Materials Sciences, Composites, Porous Materials, Physics, Mechanics and Fluid Dynamics
Journal RSS Feed