34
2. Hydrodynamic Dispersion in Porous Media
coefficient was treated as a scalar, as the relationship between dispersion and
flow direction was ignored. In fact, even in an isotropie porous medium the
dispersions in the flow and the cross sectional directions are different. In an
anisotropic medium, it becomes even more complicated.
A vast number of experiments have been done to determine the relationships between the dispersion coefficient and velocity distribution, as weIl as
the molecular diffusion coefficient. Early experiments were mainly concerned
with the longitudinal dispersion coefficient, D L • In one experiment, a fluid
with a constant tracer concentration is introduced at one end of a sand
column. The concentration of the eIDuent is measured and compared with
the analytic solution of a dispersion model. The longitudinal dispersion coefficient, D L , is then obtained from this comparison. It can be proved by
dimensional analysis that the dimensionless number DdD d is the function of
another dimensionless number
Vd
Pe=D/
(2.5.1)
where Pe is caIled the Peclet number. V is the mean pore velocity in the
porous medium and d is a characteristie length of the medium, such as
particle diameter. D d is the diffusion coefficient in solution. A large number of
experiments resulted in curves such as those shown in Figure 2.11. The curve
may be roughly divided into five zones.
Zone I. In this zone, molecular diffusion plays the predominant role. DdD d
is nearly a constant, that is, D L = D d T (because the flow velocity is very low,
the mechanical dispersion can be ignored). Proportionality coefficient T is
actuaIly the tortuosity of the medium, and thus is always less than unity,
because the porous medium retards the progress of molecular diffusion in the
liquid phase. With the zero mean flow velocity experiment (q = 0), both D L
-
-10 2
I
TI
m
IV
v
FIGURE 2.11. Relationship between longitudinal dispersion and Peclet number.
2. Hydrodynamic Dispersion in Porous Media
coefficient was treated as a scalar, as the relationship between dispersion and
flow direction was ignored. In fact, even in an isotropie porous medium the
dispersions in the flow and the cross sectional directions are different. In an
anisotropic medium, it becomes even more complicated.
A vast number of experiments have been done to determine the relationships between the dispersion coefficient and velocity distribution, as weIl as
the molecular diffusion coefficient. Early experiments were mainly concerned
with the longitudinal dispersion coefficient, D L • In one experiment, a fluid
with a constant tracer concentration is introduced at one end of a sand
column. The concentration of the eIDuent is measured and compared with
the analytic solution of a dispersion model. The longitudinal dispersion coefficient, D L , is then obtained from this comparison. It can be proved by
dimensional analysis that the dimensionless number DdD d is the function of
another dimensionless number
Vd
Pe=D/
(2.5.1)
where Pe is caIled the Peclet number. V is the mean pore velocity in the
porous medium and d is a characteristie length of the medium, such as
particle diameter. D d is the diffusion coefficient in solution. A large number of
experiments resulted in curves such as those shown in Figure 2.11. The curve
may be roughly divided into five zones.
Zone I. In this zone, molecular diffusion plays the predominant role. DdD d
is nearly a constant, that is, D L = D d T (because the flow velocity is very low,
the mechanical dispersion can be ignored). Proportionality coefficient T is
actuaIly the tortuosity of the medium, and thus is always less than unity,
because the porous medium retards the progress of molecular diffusion in the
liquid phase. With the zero mean flow velocity experiment (q = 0), both D L
-
-10 2
I
TI
m
IV
v
FIGURE 2.11. Relationship between longitudinal dispersion and Peclet number.