Page 49 - Percolation Models for Transport in Porous Media With

Chapter 3

Percolation Model of Fluid

Flow in Heterogeneous

Media

Numerous cases of violation of the linear Darcy's law, which claims that the value
of flux (flow rate) is directly proportional to the pressure gradient, has been ob-
served. For very small flow velocities (pressure gradients) these deviations are
caused by the formation of bounded fluid layers on the pore (capilliary) sur-
faces [50, 51]. For relatively large velocities, deviations are caused by turbulence
in the flow in pores (capillaries) and by the fluid kinetic energy losses on hetero-
geneities like capillary junctions, etc. [51, 52]. For sufficiently homogeneous media,
it is possible to average the law describing flow at the micro level over the whole
volume of the medium to extend it to the macro level [47]. However for some
heterogeneous media, such an operation is not valid [52]. Indeed, if r 1 and r 2 , the
radii of two successive capillaries, differ by an order of magnitude, then the local
pressure gradients in them (for example, in the case of Poiseuille flow through the
capillaries) are to one another as (rtfr2 ) 4 , i.e., differ by a factor of 10 4 • In such
a medium, all types of flow can take place at the micro level, namely the flow
with the larger part of the pore space filled with bounded fluid; Poiseuille flow;
transient flow from laminar to turbulent; and turbulent flow.

3.1 Flow at the Micro Level

Begin with some preliminary remarks and introduce some notations which will
make the further presentation easier.

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