Abstract
An average streamwise channel velocity is proposed as a more accurate representation of the actual intrapore velocity than the intrinsic phase average velocity. A relationship is derived between the average streamwise channel velocity and the interstitial velocity and superficial velocity. New definitions of tortuosity and areosity as second-order tensors are proposed for porous media in general. Novel names, semantically in line with the respective physical meanings, are proposed for these quantities. The definitions produce results which conform with several other published results and are applicable to anisotropic media in general.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- â o :
-
total area of representative elementary area
- â of :
-
fluid phase ofâ o
- \(\hat A_{o\mathcal{L}}\) :
-
effective streamwise part ofâ of
- A ij :
-
areosity tensor
- c :
-
molar concentration of a chemical species
- \(\mathop c\limits^o\) :
-
deviation of molar concentration,\(( = c - 1/\mathcal{U}_{of} \int_{\mathcal{U}of} {cdV)}\),
- ε :
-
electrical tortuosity factor
- F :
-
formation factor
- \(\hat L\) :
-
average streamwise displacement
- \(\hat L_e\) :
-
average channel length inU o
- \(\mathcal{L}_{ij}\) :
-
lineality tensor
- \(\hat Q_i\) :
-
total discharge through REV
- \(\hat q\) :
-
magnitude of\(\hat q_i\),
- \(\hat q_i\) :
-
specific discharge, superficial velocity
- S :
-
surface
- S f f :
-
fluid-fluid phase contact overS of
- S f s :
-
fluid-solid phase contact overS of
- S of :
-
total boundary of fluid phase
- σ:
-
curve length along streamline
- U :
-
volume
- U o :
-
volume of REV
- U of :
-
fluid phase within REV
- \(\hat U_{fL}\) :
-
effective streamwise part ofU of ,
- û i :
-
macroscopic intrinsic phase average velocity, interstitial velocity
- \(\tilde \upsilon _i\) :
-
microscopic fluid phase velocity withinU of ,
- w :
-
intrapore speed in capillary tube
- ŵ :
-
magnitude ofŵ i
- \(\tilde w\) :
-
magnitude of\(\tilde w_i\),
- ŵ i :
-
average streamwise channel velocity withinU of ,
- \(\tilde w_i\) :
-
channel average velocity withinU of ,
- x i :
-
i coordinate
- x I :
-
I principal coordinate axis
- ¯x i :
-
position vector indicating centroid of REV
- ¯x f i :
-
position vector indicating centroid of fluid volume within REV
- \(\mathop x\limits^o _i\) :
-
position vector relative to REV centroid
- δ ij :
-
Kronecker delta
- ε :
-
volume fraction of fluid phase inU o , porosity
- ε A :
-
areal fraction of fluid phase inâ o , porosity
- Ν i :
-
outward directed unit normal vector on surface
- \(\tilde v_i\) :
-
unit vector tangent to streamline
- \(\hat v_i\) :
-
streamwise unit vector
- \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{v} _i\) :
-
unit vector in plane orthogonal to\(\hat v_i\),
- Τ i :
-
tortuosity vector
- χ ij :
-
tortuosity tensor
- ξ I :
-
principal areosity value
- I :
-
as subscript, pertaining to principal direction of tensor
- λ:
-
pertaining to streamwise direction
- ∼:
-
pertaining to local flow direction
References
Bear, J.: 1972,Dynamics of Fluids in Porous Media, American Elsevier, New York.
Bear, J. and Bachmat, Y.: 1986, Macroscopic modelling of transport phenomena in porous media, 2: Applications to mass, momentum and energy transport,Transport in Porous Media,1, 241–269.
Bear, J. and Bachmat, Y.: 1991,Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer, Dordrecht.
Carman, P. C.: 1937, Fluid flow through granular beds,Trans. Inst. Chem. Eng. 15, 150–158.
Carman, P. C., 1956,Flow of Gases through Porous Media, Butterworth, London.
Comiti, J. and Renaud, M.: 1989, A new model for determining mean structure parameters of fixed beds from pressure drop measurements: applications to beds packed with parallelepiped particles,Chem. Eng. Sci. 44 (7), 1539–1545.
Dagan, G.: 1989,Flow and Transport in Porous Formations, Springer, Berlin.
Du Plessis, J. P. and Masliyah, J. H.: 1988 Mathematical modelling of flow through consolidated isotropic porous media,Transport in Porous Media,3, 145–161.
Du Plessis, J. P., 1992, Pore-scale modelling for flow through different types of porous environments, in Quintard, M. and Todorvic, M. (eds),Heat and Mass Transfer in Porous Media, Elsevier, Amsterdam, pp. 281–292.
Dullien, F. A. L.: 1979,Porous Media Fluid Transport and Pore Structure, Academic Press, New York.
Gray, W. G.: 1975, A derivation of the equations for multi-phase transport,Chem. Eng. Sci. 30 (2), 229–233.
Kemblowski, Z. and Michniewicz, M.: 1979, A new look at the laminar flow of power law fluids through granular beds,Rheolog. Acta,18, 730–739.
Ruth, D. and Ma, H.: 1992, On the derivation of the Forchheimer equation by means of the average theorem,Transport in Porous Media,7 (3), 255–264.
Scheidegger, A. E.: 1972,The Physics of Flow through Porous Media, University of Toronto Press.
Slattery, J. C.: 1972,Momentum, Energy and Mass Transfer in Continua, McGraw-Hill, New York.
Spearing, M. and Matthews, P.: 1991, Modelling characteristic properties of sandstones,Transport in Porous Media 6, 71–90.
Suman, R. and Ruth, D.: 1993, Formation factor and tortuosity of homogeneous porous media,Transport in Porous Media 7, 185–206.
Whitaker, S., 1967, Diffusion and Dispersion in Porous Media,A.I.Ch.E. J. 13, 420–427.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Diedericks, G.P.J., Du Plessis, J.P. On tortuosity and areosity tensors for porous media. Transp Porous Med 20, 265–279 (1995). https://doi.org/10.1007/BF01073176
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01073176