(1973) n°3 - Royal Academy for Overseas Sciences

— 587 —
PoREH and El a t a ( 2 3 ) . However, one still needed to know the
detailed structure of the medium geometry to deduce the unknown
coefficient of proportionality i.e. the permeability tensor.
For the flow of a homogeneous incompressible fluid through
an isotropic homogeneous porous medium. Darcy’s law is
— v p — p g k — p ^ U = 0, (1)
K.
where u is the mean velocity or the average flux through an extensive
porous medium per unit surface, p is the pressure, v the
viscosity, p the density, K the permeability, and v is the del-
operator. The gravity acceleration is g, and k is a unit vector
directed upwards.
The issue of the validity of equation ( l) is further obscured
by the similarity of the average flux relation for steady Poi-
seuille flow through a straight pipe, and for the average flux
through an extensive porous medium. In both cases, it turns out
that the volume flux is proportional to the pressure gradient.
This similarity has tended to mask the very important conceptual
differences between the two situations.
For the flow through an anisotropic porous medium, it is
generally assumed that equation ( l) is valid provided that one
uses a directional permeability tensor (Ka) O ’B r ie n (21)
has performed viscous flow calculations for some idealized orthotropic
porous models. The calculations provided some basis
for quantitative values of a directional permeability tensor to
replace omni-directional and semi-empirical engineering estimates.
A generalized Darcy’s law was analytically derived by S a f f -
m a n (26c) for a homogeneous fluid. He defined two quantities
which he averaged over the porous medium surface: the interstitial
velocity q(x), a continuous function which vanishes at
the solid surfaces of the matrix, and the interstitial pressure
p*(x), a function that is discontinuous over the solid surfaces
of the medium. For steady inertia-free flows, he writes the
generalized Darcy’s law as
pvKa •u = p v v : u — ?;V p — Tjpgk (2)
where i? is the porosity. This equation still contains the Lapla-