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Transport in Porous Media

, Volume 3, Issue 2, pp 145–161 | Cite as

Mathematical modelling of flow through consolidated isotropic porous media

  • J. Prieur Du Plessis
  • Jacob H. Masliyah
Article

Abstract

A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.

Key words

Porous medium REV averaging inertial effects tortuosity isotropic square duct 

Notation

Ap

pore cross-sectional area

d

microscopic characteristic length

de

flow length within RUC

dp

pore width

f

friction factor,\(\left( {\frac{{\partial p}}{{\partial x}} \cdot 2d_p } \right)/(p\upsilon _p^2 ),\)

fapp

friction factor, (2dpΔp)/(γν 2 p Δx)

g

gravity constant

I

integral expression

K

hydrodynamic permeability

lf

frictional flow length within RUC

n

porosity

p

pressure

q

specific discharge (= 〈v〉)

Re

Reynolds number,γυpdp/μ

Reqd

Reynolds number,γqd/μ

S

surface

Sfs

fluid-solid interface within RUC or REV

T

tortuosity

V

volume

v

fluid velocity withinVn

vn

average fluid velocity withinvn

vp

mean fluid velocity within pore section

x

axial distance

x+

dimensionless axial distance

β

inertia parameter

μ

fluid dynamic viscosity

μ′

Brinkman effective viscosity

ν

normal vector onSfs

ϱ

fluid density

φ

extensive tensorial property

c

as subscript, denotes central or critical value

i

as subscript, denotes inflection point

n

as subscript, pertaining to void volume

o

as subscript, denotes total volume of RUC or REV

p

as subscript, pertaining to pore section

μ

a subscript, denotes streamwise shear term

vector operator ‘del’

(·)

deviation of ( ) from 〈( )〉 n , tensor inner product

〈( )〉

volumetric phase average of ( )

〈( )〉n

volumetric intrinsic phase average of ( )

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Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • J. Prieur Du Plessis
    • 1
  • Jacob H. Masliyah
    • 2
  1. 1.Dept. of Mechanical EngineeringUniversity of PretoriaPretoriaSouth Africa
  2. 2.Dept. of Chemical EngineeringUniversity of AlbertaEdmontonCanada

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