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Poiseuille-Number-Based Kozeny–Carman Model for Computation of Pore Shape Factors on Arbitrary Cross Sections

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Abstract

Porous media characterization is crucial to engineering projects where the pore shape has impact on performance gains. Membrane filters, sportswear fabrics, and tertiary oil recovery are a few examples. Kozeny–Carman (K–C) models are one of the most frequently used to understand, for instance, the relation between porosity, permeability, and other small-scale parameters. However, they have limitations, such as the inability to capture the correct dependence of permeability on porosity, the imperfect handling of the linear and nonlinear effects yielded by its fundamental quantities, and the insufficiency of geometrical parameters to predict the permeability correctly. In this paper, we cope with the problem of determining shape factors for generic geometries that represent sundry porous media configurations. Specifically, we propose a method that embeds the Poiseuille number into the classical K–C equation and returns a substitute shape factor term for its original counterpart. To the best of our knowledge, the existing formulations are unable to obtain shape factors for pores whose geometry is beyond the regular ones. We apply a Galerkin-based integral (GBI) method that determines shape factors for generic cross sections of pore channels. The approach is tested on straight capillaries with arbitrary cross sections subject to steady single-phase flow under the laminar regime. We show that shape factors for basic geometries known from experimental results are replicable exactly. Besides, we provide shape factors with precision up to 4 digits for a class of geometries of interest. As a way to demonstrate the applicability of the GBI approach, we report a case study that determines shape factors for 19 generic individual pore sections of a laboratory experiment involving flow rate measurements in an industrial arrangement of a water-agar packed bed. Porosity, flow behavior, and velocity distributions determined numerically achieve a narrow agreement with experimental values. The findings of this study provide parameters that can help to design new devices or mechanisms that depend on arbitrary pore shapes, as well as to characterize fluid flows in heterogeneous porous media.

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Data Availability

Any data or material will be available by the authors under request.

Code Availability

The code may be available by the authors under request.

Notes

  1. Moved to appendix due to its length.

  2. OpenCV (Open Source Computer Vision Library) is an open-source computer vision and machine learning software library freely available on https://opencv.org.

  3. Although (Berger and Zhou 2014) calls it Smirnov test only, we will maintain the most-known nomenclature.

Abbreviations

\(\alpha ,\beta\) :

Geometric parameters

\({\bar{\tau }}_w\) :

Mean shear stress [Pa]

\(\epsilon ,r,n\) :

Geometric parameters

\(\varGamma\) :

Duct’s boundary

\(\varGamma _p^{(i)}\) :

Individual pore boundary

\({\mathcal {P}}\) :

Distribution [dimensionless]

\(\mu\) :

Fluid viscosity [Pa s]

\(\varOmega _p^{(i)}\) :

Individual pore domain

\(\phi\) :

Porosity [dimensionless]

\(\phi _e\) :

Effective porosity [dimensionless]

\(\rho\) :

Density [\(\hbox {kg/m}^{3}\)]

\(\tau\) :

Tortuosity [dimensionless]

a, b :

geometric parameters

\(A_c\) :

Cross-sectional area of the representative volume [\(\hbox {m}^{2}\)]

\(A_p\) :

Cross-sectional porous area [\(\hbox {m}^{2}\)]

\(A_s\) :

Surface area [\(\hbox {m}^{2}\)]

\(C_H\) :

Hazen’s coefficient

D :

Statistics D-value

d :

Particle size [mm]

\(D_h\) :

Dimensionless hydraulic diameter

\(d_h\) :

Hydraulic diameter [m]

dp/dz :

Pressure gradient [Pa/m]

f :

Fanning’s friction factor [dimensionless]

\(F_{KC}\) :

Kozeny–Carman shape factor [dimensionless]

\(F_{val}\) :

Shape factor [dimensionless]

K :

Absolute permeability [\(\upmu \hbox {m}^{2}\)]

L :

Virtual (straight) length [m]

l :

Characteristic length [m]

\(L_g\) :

True (geodesic) length [m]

P :

Perimeter [m]

Po :

Poiseuille number [dimensionless]

q :

Volume flow rate [\(\hbox {m}^{3}/\hbox {s}\)]

Re :

Reynolds number [dimensionless]

\(S_{Vgr}\) :

Grain’s specific surface area [\(\upmu \hbox {m}^{-1}\)]

u :

Local fluid velocity [m/s]

\(u_m\) :

Mean velocity [m/s]

V :

Velocity field [\(\hbox {cm s}^{-1}\)]

\(V_p\) :

Pore volume [\(\hbox {m}^{3}\)]

\(V_t\) :

Total volume [\(\hbox {m}^{3}\)]

\(V_{gr}\) :

Grain volume [\(\hbox {m}^{3}\)]

W :

Local dimensionless velocity

\(W/W_m\) :

Dimensionless velocity field

\(W_m\) :

Dimensionless mean velocity

X, Y, Z :

Dimensionless coordinates

x, y, z :

Cartesian coordinates [m]

REV:

Representative elementary volume

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Acknowledgements

G.P.O. thanks Túlio Souza from Federal University of Paraíba for technical discussions and support concerning OpenCV image processing.

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Authors and Affiliations

  1. Department of Physics and Civil Engineering, State University of Paraíba, Center of Science, Technology and Health, Araruna, PB, Brazil

    Valdecir A. Santos Júnior & Antônio F. Silva Júnior

  2. Petroleum Engineering Modelling Laboratory, Federal University of Paraíba, João Pessoa, PB, Brazil

    Tatiana A. Simões

  3. Postgraduate Program in Mechanical Engineering, Federal University of Paraíba, João Pessoa, PB, Brazil

    Gustavo P. Oliveira

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  1. Valdecir A. Santos Júnior
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  2. Antônio F. Silva Júnior
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  3. Tatiana A. Simões
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  4. Gustavo P. Oliveira
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Contributions

Find below the CRediT taxonomy for this manuscript: VASJ was involved in conceptualization. Methodology, formal analysis and investigation, writing—original draft preparation, and supervision. AFSJ helped in resources. TAS contributed to formal analysis and investigation, resources. GPO was involved in formal analysis and investigation, resources, writing—original draft preparation, and writing—review and editing.

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Correspondence to Gustavo P. Oliveira.

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Appendix

Appendix

1.1 Overview on Mathematical Models Derived from Kozeny–Carman’s Equation

See Tables 3, 4 and 5.

Table 3 A class of modified Kozeny–Carman equations proposed for different porous media applications. Adapted from Xu and Yu (2008)
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Table 4 Multiple definitions for shape factor found in the literature over the last 60 years
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Table 5 Literature studies on the determination of Kozeny’s constant. In the second column, T, E, and C stand for “Theoretical,” “Experimental,” and “Computational,” in this order. Adapted from Ozgumus et al. (2014)
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1.2 Shape Factors for Geometries with Arbitrary Cross-Section

See Table 6.

Table 6 Shape factors computed from \(F_{val}\) expression for a class of geometries of interest with arbitrary cross sections. Compiled from Shah and London (1978)
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Júnior, V.A.S., Júnior, A.F.S., Simões, T.A. et al. Poiseuille-Number-Based Kozeny–Carman Model for Computation of Pore Shape Factors on Arbitrary Cross Sections. Transp Porous Med 138, 99–131 (2021). https://doi.org/10.1007/s11242-021-01592-4

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