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Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach

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Abstract

This is the first of two papers presenting a systematic development of a continuum model of a porous medium and of transport processes occurring in it. The concept of a Representative Elementary Volume (REV) as opposed to any arbitrary volume of averaging quantities at the micro-scale, is quantified. A universal criterion for selecting the size of an REV as a function of measurable characteristics of a porous medium and selected tolerance levels of estimation errors, is developed. The rules of spatial averaging are extended by including the effects of both the configuration of the solid matrix and of interphase transfer phenomena within an REV.

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Abbreviations

a o :

a constant (also a 1, a 2, a 3)

A :

area

A o :

Representative Elementary Area (REA)

b :

length of side of cubical box

b :

grad n¦ x o

\(C'_\alpha \) :

empirical numerical coefficient

C Δ :

a coefficient defined in (15)

(D):

domain

e :

density of E

E :

extensive quantity

G ijk :

a tensorial property

G α :

property of α-phase

h:

oriented distance

J E α :

e α V E α (flux of E of α), diameter of REV

l :

diameter of U 0 (also l min, l max)

L * :

characteristic length of domain

n :

porosity

N :

number of items

R :

radius of REV

s :

length

(S):

surface of area S

(S αβ ):

interface between a and α-phases within REV

(S 0):

surface bounding (U 0)

(S αα ):

α-α portion of (S 0)

(S 0α ):

total surface bounding (U 0α )

t :

time

T * α :

coefficient related to the geometrical configuration of α phase within REV

u :

velocity of displacement of (S αβ )

u 0 :

volume of representative elementary volume (REV)

(U):

domain of volume U

(U 0α ):

domain occupied by α-phase within (U 0)

V :

velocity vector

V E α :

velocity of E α -continuum

x :

position vector of point

x 0 :

position vector of centroid of REV

\(\mathop x\limits^ \circ\) :

x - x 0

β:

probability

γ α (X):

characteristic function of α-phase

λ(x):

characteristic function of void space

Δ:

hydraulic radius

δ:

arbitrary number in the range (0,1)

ε:

error level

θ α :

volumetric fraction of α-phase (=U 0α /U 0)

θ A α :

areal fraction (=A 0α /A 0)

θ S α :

fraction of a on S 0

κ :

empirical coefficient

λ α :

coefficient of the α-phase

ν α :

unit normal vector, outward to (U 0α ) on, (S 0α )

π α :

quantity associated with area of α-phase

\(\sigma _{\hat n}^2\) :

variance of \(\hat n\)

τ γ (h):

correlation coefficient defined in (7)

ρ α :

mass density of α-phase

α :

as subscript, denotes α-phase

ν :

as subscript, denotes void space (0ν = void space in REV)

s :

as subscript, denotes solid (0s = solid in REV)

0:

as subscript denotes ‘of REV’

(°):

deviation of ( ) from its average \(\overline {( )} ^\alpha\)

E( ):

expected value of ( )

\(\overline {( )}\) :

average (=(1/U)∫(U)( ) d U); volumetric phase average (=(1/U 0)∫(U )0α ( ) d U α )

\(\overline {( )} ^\alpha\) :

volumetric intrinsic phase average (=(1/U)0α ∫(U )0α ( ) d U α )

\(\tilde (\widetilde{}{\tilde )}^i\) :

areal average ( ) over surface normal to x i -axis

\(\tilde (\widetilde{}{\tilde )}^{\alpha \beta }\) :

average of ( ) over (S αβ )

〈( )〉:

areal average of ( )

〈( )〉 α :

areal intrinsic phase average of ( )

〈( )〉s 0 :

average of ( ) along s 0

[( )]1,2 :

( )¦1 - ( )¦2 = jump in ( ) between sides 1 and 2 of a surface

\(()\) :

estimate of ( )

DE α ( )/Dt :

material derivative of ( )

1x i :

unit vector in +x i -direction

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Author information

Authors and Affiliations

  1. Hydrological Service, Jerusalem, Israel

    Yehuda Bachmat (Director)

  2. Technion-Israel Institute of Technology, Haifa, Israel

    Jacob Bear (Albert and Anne Mansfield Professor of Water Resources)

Authors
  1. Yehuda Bachmat
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  2. Jacob Bear
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Bachmat, Y., Bear, J. Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach. Transp Porous Med 1, 213–240 (1986). https://doi.org/10.1007/BF00238181

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  • DOI: https://doi.org/10.1007/BF00238181

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