Page 53 - Percolation Models for Transport in Porous Media With

3.1 FLOW AT THE MICRO LEVEL 45

Laminar flow (Re < Rel). Deviations from the Darcy's law for very small
values of the flow rate are usually explained by the formation of the bounded fluid
layers on the surfaces of pores (capillaries). These layers can fill a good part of
the pore space and can decrease the permeability of the medium significantly, up
to the total termination of the flow [50].
Results of a series of experiments on the flow in homogeneous media under
small pressure gradients can be found in [54]. Present the empirical dependence
that describes these results [50]

(3.12)

Here Go is the minimal pressure gradient, when the flow begins; Ko is the
permeability for pressure gradients G » G0 , where Darcy's law is valid. Assuming
that the pore space of a homogeneous medium consists of capillaries with radii r
close to the average radius, after substitutions

in (3.12) and some transformations, we can obtain the following correlation be-
tween the pressure gradient in the capillary and the value of flow in it

(3.13)

Here g0 ( r) has the meaning of the minimal pressure gradient in a capillary.
When qf » r 4 go(r) we get the well-known Poiseuille's formula

Transient regime of flow (Re1 :$Re:$ Re2). The flow in this domain is inter-
mediate from laminar and turbulent (Nikuradze's saddle [12]). It does not depend
on the roughness of the capillary and can be described by a certain approximate
formula. The latter is obtained after studying the plots of hydraulic resistance
against the Reynolds number

(3.14)

Here i = 2 + 4, j = 4 + 8, Km is a dimensioned constant which depends on PI
and JL.
Turbulent flow (Re > Re2). For a rough capillary, the following approximate
formula can be used
(3.15)
Here s = 1.8 + 2.5; h = 4 + 6, Kt is a dimensioned constant which depends on
p f, JL, and the degree and the type of roughness.