2
Hydrodynamic Dispersion in
Porous Media
2.1 Physical Parameters
2.1.1 Spatial Average M ethod
When fluid flows through the interconnected voids and passages of a porous
medium, the walls of these voids and passages form many small tunnels, and
the fluid flows inside them. The study of physical phenomena in a porous
medium on such ascale (pore scale) is called the microscopic method. Due to
the complexity of the micro-geometry of porous media, it is unrealistic to
study the details of the flow on this scale. Therefore, one has to describe the
flow phenomena in porous media on a macroscopic rather than microseopie
basis. The spatial average method is a way to transfer properties of porous
media from the microscopic level to the macroscopic level.
Consider a mathematical point x in a flow region, with coordinates
(Xl' X2' x3 ) in a three-dimensional coordinate system. A small volume,
[Uo(x)], which can be either spheric or cubic with its center at x, is defined as
a particle of the porous medium. On the one hand, the volume [Uo(x)] must
be large enough to cover a sufficient number of solid particles and pores so
that the stable mean values of certain physical properties associated with
[Uo(x)] can be obtained over the volume. For example, if the pore space of
[Uo(x)] is marked as [Uo,v(x)], and [Uo(x)] varies within a certain range,
then the ratio of volume
n(x) = Uo,v(x)
Uo(x)
(2.1.1)
may be nearly a constant. As a result, it can be defined as the porosity at point
x. On the other hand, the volume of [Uo(x)] must also be small enough in
comparison with the whole region so that it can be treated as a point. The
particle thus defined is also called a Representative Elementary Volume (REV)
of porous media (Bear, 1972).
If every mathematical point in a porous medium is associated with a
particle, then the porous medium, which is constructed with solid matrices
9
Hydrodynamic Dispersion in
Porous Media
2.1 Physical Parameters
2.1.1 Spatial Average M ethod
When fluid flows through the interconnected voids and passages of a porous
medium, the walls of these voids and passages form many small tunnels, and
the fluid flows inside them. The study of physical phenomena in a porous
medium on such ascale (pore scale) is called the microscopic method. Due to
the complexity of the micro-geometry of porous media, it is unrealistic to
study the details of the flow on this scale. Therefore, one has to describe the
flow phenomena in porous media on a macroscopic rather than microseopie
basis. The spatial average method is a way to transfer properties of porous
media from the microscopic level to the macroscopic level.
Consider a mathematical point x in a flow region, with coordinates
(Xl' X2' x3 ) in a three-dimensional coordinate system. A small volume,
[Uo(x)], which can be either spheric or cubic with its center at x, is defined as
a particle of the porous medium. On the one hand, the volume [Uo(x)] must
be large enough to cover a sufficient number of solid particles and pores so
that the stable mean values of certain physical properties associated with
[Uo(x)] can be obtained over the volume. For example, if the pore space of
[Uo(x)] is marked as [Uo,v(x)], and [Uo(x)] varies within a certain range,
then the ratio of volume
n(x) = Uo,v(x)
Uo(x)
(2.1.1)
may be nearly a constant. As a result, it can be defined as the porosity at point
x. On the other hand, the volume of [Uo(x)] must also be small enough in
comparison with the whole region so that it can be treated as a point. The
particle thus defined is also called a Representative Elementary Volume (REV)
of porous media (Bear, 1972).
If every mathematical point in a porous medium is associated with a
particle, then the porous medium, which is constructed with solid matrices
9