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

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dan of the velocity vector. Its inclusion in Darcy’s law is
associated with the name of B r i n k m a n . Furthermore, the interstitial
velocity and the force that the viscous fluid exerts on
the porous medium, were shown to be correlated by a simple
integral equation whose kernel is precisely the permeability
tensor.
3b. Assumptions for inhomogeneous systems.
W o o d i n g (36d) concluded that Darcy’s law ( l ) is applicable
to the flow, unidirectional in the mean, of a homogeneous
fluid through a porous medium spatially homogeneous in its
“average properties”. W o o d i n g further stated that when either
or both conditions is not satisfied, the validity of equation ( l)
can be tested by considering the relations between the characte-
ristric length scales involved in the system considered. It was
argued by G e l h a r (8) that Darcy’s law is locally valid in the
case of a medium whose properties fluctuate randomly, provided
that the scale of the random fluctuations is large compared
to a typical grain size of the material constituting the medium.
This grain size is given the name d. If the moving fluid shows
spatial variations in viscosity and (or) density, the validity of
Darcy’s law is assumed implicit in a system where d is much
smaller than any characteristic dimension of the macroscopic
flow. W hen studying the motion of a density-stratified fluid,
an additional gravitational driving head entering the equations
of motion is the buoyancy b,
b = g AP / f S
w h ere Ap is a characteristic d en sity d iffe ren ce or th e departure
o f th e d en sity p from a referen ce d en sity po. T h e prim ary flo w
is characterized by a rate o f flo w A,
A. = K g A p/po V .
By similarity, the characteristic rate of secondary flow can be
written as (36d)
A fKg j V dyp\ Ap_
I V v d p J po
where V is the characteristic primary volume flow rate. These