14
2. Hydrodynamic Dispersion in Porous Media
All Newtonian fluids obey the following law:
ou
r = Jl. on '
(2.1.20)
where r is the shear stress; au/an, the velocity gradient normal to liquid
surface; and Jl., a coefficient which is called the dynamic viscosity. The ratio of
dynamic viscosity to density is called the kinematic viscosity, v = Jl./ p. All of
these properties can be transferred to their macroscopic values by the spatial
average method.
Note that the density and viscosity of a solution will change with press ure
p, solute concentration C and temperature T. At a constant temperature,
their relationship can be expressed as
p = p(C,p),
Jl. = Jl.(C,p),
(2.1.21)
which are called state equations. As a first approximation, density and viscosity can be regarded as linear functions of C and p.
Flow Velocity of Multicomponent Fluids
Let vector V /Z be the flow velocity of component IX on a microscopic level,
then two macroscopic mean velocities can be defined for this fluid system, i.e.,
the mean mass velocity
N
V= L W/Z V/Z ,
(2.1.22)
«=1
and the mean volume velocity
(2.1.23)
The former uses mass fractions as weighting coefficients, and the latter uses
volume fractions. We will mainly use the mean mass velocity throughout this
book. The mean volume velocity is used only when the volume of fluid mixture is dependent on the densities of the components.
The relevant macroscopic averages of Eq. (2.1.22) and Eq. (2.1.23) can be
obtained by spatial averaging. The averages of V and V' in a REV of porous
media are defined as
-
1 i
V = - -
VdUop,
Uo,p(x) [Uo.,(x)]
,
(2.1.24)
V' = _1_ r V' dUo p,
Uo,p(x) JIUo.,(X)]
•
(2.1.25)
where [Uo,p(x)] is the volume occupied by a multicomponent fluid in the
REV. If a porous medium is saturated with an incompressible homogeneous
2. Hydrodynamic Dispersion in Porous Media
All Newtonian fluids obey the following law:
ou
r = Jl. on '
(2.1.20)
where r is the shear stress; au/an, the velocity gradient normal to liquid
surface; and Jl., a coefficient which is called the dynamic viscosity. The ratio of
dynamic viscosity to density is called the kinematic viscosity, v = Jl./ p. All of
these properties can be transferred to their macroscopic values by the spatial
average method.
Note that the density and viscosity of a solution will change with press ure
p, solute concentration C and temperature T. At a constant temperature,
their relationship can be expressed as
p = p(C,p),
Jl. = Jl.(C,p),
(2.1.21)
which are called state equations. As a first approximation, density and viscosity can be regarded as linear functions of C and p.
Flow Velocity of Multicomponent Fluids
Let vector V /Z be the flow velocity of component IX on a microscopic level,
then two macroscopic mean velocities can be defined for this fluid system, i.e.,
the mean mass velocity
N
V= L W/Z V/Z ,
(2.1.22)
«=1
and the mean volume velocity
(2.1.23)
The former uses mass fractions as weighting coefficients, and the latter uses
volume fractions. We will mainly use the mean mass velocity throughout this
book. The mean volume velocity is used only when the volume of fluid mixture is dependent on the densities of the components.
The relevant macroscopic averages of Eq. (2.1.22) and Eq. (2.1.23) can be
obtained by spatial averaging. The averages of V and V' in a REV of porous
media are defined as
-
1 i
V = - -
VdUop,
Uo,p(x) [Uo.,(x)]
,
(2.1.24)
V' = _1_ r V' dUo p,
Uo,p(x) JIUo.,(X)]
•
(2.1.25)
where [Uo,p(x)] is the volume occupied by a multicomponent fluid in the
REV. If a porous medium is saturated with an incompressible homogeneous