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

Bartłomiej Bodak; Maciej Sobótka

doi:10.2478/sgem-2024-0001

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

Studia Geotechnica et Mechanica

Volume 46 (2024): Issue 1 (March 2024)

By: Bartłomiej Bodak and Maciej Sobótka

Open Access

|Mar 2024

Figures & Tables

Grain size distribution curves of analyzed samples.

General concept of a pore-network model.

Analogous model of the resistor network.

Rendered view of reconstructed 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.

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.

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.

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

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

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

Streamlines of flow calculated using LBM: 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]	φ_img [−]	k [μm²]	K [m/s]
1	Fine sand	400³	0.365	23.489	1.758E-4
1	Fine sand	600³	0.364	17.567	1.317E-4
2	Fine sand with lignite	400³	0.511	20.923	1.565E-4
2	Fine sand with lignite	600³	0.511	23.193	1.736E-4
3	Medium sand	400³	0.309	16.778	1.259E-4
3	Medium sand	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

		K_equiv [m/s]	K_ap [m/s]	K_ex [m/s]	K_corr [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]	φ_img [−]	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]	ρ_s [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]*	*φ_img [−]*	*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 d_e	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*
Hazen (1911)	(4)	6.0E-4	1+10(φ−0.26)	d₁₀	2	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
Beyer (1964)	(4)	6E-4·log(500/U)	1	d₁₀	2	1<U<20
Chapuis et al. (2005)	(5)	1219.9	φ^2.3475/(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

		*[−]*	*[−]*	*d_e [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

Articles in this issue

DOI: https://doi.org/10.2478/sgem-2024-0001 | Journal eISSN: 2083-831X | Journal ISSN: 0137-6365

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Language: English

Page range: 1 - 20

Published on: Mar 29, 2024

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

microtomography,

permeability,

conductivity,

computational fluid dynamics,

Kozeny–Carman equation

Related subjects:

Physics,

Mechanics and fluid dynamics

© 2024 Bartłomiej Bodak, Maciej Sobótka, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 46 (2024): Issue 1 (March 2024)

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

Figures & Tables

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Results of simulations using the lattice-Boltzmann method_

Results of measurements with the described small-scale permeameter setup_

Results of simulations using the pore-network modeling approach_

Measured properties of the samples_

Results of estimation using the Kozeny–Carman equation_

Summary of used empirical formulae_

Results of estimation using empirical equations_

Paradigm

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