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

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

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 nameVOI sizePorosity derived from image dataPermeabilityHydraulic conductivity at 10°C
[vx]φimg [−]k [μm2]K [m/s]
1Fine sand40030.36523.4891.758E-4
60030.36417.5671.317E-4
2Fine sand with lignite40030.51120.9231.565E-4
60030.51123.1931.736E-4
3Medium sand40030.30916.7781.259E-4
60030.31715.3961.151E-4

Results of measurements with the described small-scale permeameter setup_

Sample no.Sample nameMean conductivity derived from the best-fit curveConductivity of the apparatusHydraulic conductivity in the measurement temperatureHydraulic conductivity at 10°C
Kequiv [m/s]Kap [m/s]Kex [m/s]Kcorr [m/s]
1Fine sand2.663E-54.927E-32.678E-51.951E-5
2Fine sand with lignite4.457E-6 4.461E-63.250E-6
3Medium sand6.183E-5 6.262E-54.562E-5

Results of simulations using the pore-network modeling approach_

Sample no.Sample nameVOI sizePorosity derived from image dataPermeabilityHydraulic conductivity at 10°C
[vx]φimg [−]k [μm2]K [m/s]
1Fine sand40030.36523.6661.786E-4
60030.36423.5871.780E-4
80030.36324.0611.816E-4
2Fine sand with lignite40030.51128.4332.145E-4
60030.51127.9692.110E-4
80030.50627.3382.063E-4
3Medium sand40030.30917.3111.306E-4
60030.31720.3011.532E-4
80030.31722.0871.667E-4

Measured properties of the samples_

Sample no.Sample nameSoil type according to PN-EN ISO 14688-2:2018Bulk densitySpecific densityPorosity in loose stateHydraulic conductivity in falling-head test at 10°CUniformity coefficient U=d60/d10GSD curve slope coefficient C=d302/(d60·d10)
[−]ρ [g/cm3]ρs [g/cm3]φ [−]K [m/s]U [−]C [−]
1Fine sandFSa1.5492.6340.4121.702E-51.8401.054
2Fine sand with ligniteFSa1.2382.6440.5323.189E-62.5321.027
3Medium sandMSa1.6522.6540.3774.067E-53.1471.003

Results of estimation using the Kozeny–Carman equation_

Sample no.Sample nameVOI sizePorosity derived from image dataTortuosity in direction of the flowSpecific surface area per unit volumePermeabilityHydraulic conductivity at 10°C
[vx]φimg [−]τ [−]S [1/m]k [μm2]K [m/s]
1Fine sand40030.3651.9373874816.5871.242E-4
60030.3641.9353844716.6741.249E-4
80030.3631.8973782017.3781.301E-4
2Fine sand with lignite40030.5111.7327232124.6391.845E-4
60030.5111.7227306124.2831.818E-4
80030.5061.7637293222.6451.696E-4
3Medium sand40030.3092.009409887.3235.484E-4
60030.3171.980400438.6046.443E-4
80030.3171.9463707910.2097.645E-4

Summary of used empirical formulae_

MethodEquation formCoefficient C or C′Porosity function f(φ)Effective diameter deExponent mApplicability
Seelheim (1880)(5)35701d502Sands and clays
Hazen (1911)(4)6.0E-41+10(φ−0.26)d1020.1 mm<d10<3 mm*
U<5
Sauerbrey (1932)(4)3.75E-3φ3/(1−φ)2d172d17<5 mm
USBR (Říha et al., 2018)(4)4.8E-4·(1000d20)0.31d202U<5
Beyer (1964)(4)6E-4·log(500/U)1d1020.06 mm<d10<0.6 mm
1<U<20
Chapuis et al. (2005)(5)1219.9φ2.3475/(1−φ)1.565d101.5650.03 mm<d10<3 mm

Results of estimation using empirical equations_

Sample no.Sample nameMethodEffective diameterEffective diameter valueHydraulic conductivity at 10°C
[−][−]de [mm]K [m/s]
1Fine sandSeelheimd500.2732.661E-4
Hazend100.1633.031E-4
Sauerbreyd170.1892.045E-4
USBRd200.2018.937E-5
Beyerd100.1632.927E-4
Chapuisd100.1634.125E-4
2Fine sand with ligniteSeelheimd500.1386.799E-5
Hazend100.062N/A
Sauerbreyd170.0771.151E-4
USBRd200.0821.150E-5
Beyerd100.0623.994E-5
Chapuisd100.0622.363E-4
3Medium sandSeelheimd500.3815.182E-4
Hazend100.1432.01E-4
Sauerbreyd170.1791.254E-4
USBRd200.1968.531E-5
Beyerd100.1432.037E-4
Chapuisd100.1432.496E-4
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:
Geosciences,
Geosciences, other,
Materials sciences,
Composites,
Porous materials,
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.

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