2.5. Coefficients of Hydrodynamic Dispersion
37
expressions of dispersion coefficients, theoretical models have to be used.
Bear and Bachmet (1967) and Bear (1972) recommended the following expression for mechanical dispersion coefficients using their statistical geometrical model (Bear, 1972):
,
VmVn
~
Dij = IXijmn ----=- f(Pe, u),
V
(2.5.8)
where Einstein's summation convention is implied. IXijmn is a component of a
fourth rank tensor, called the dispersivity ofporous media, and has the dimension of length. V m and V n (m, n = 1,2,3) are components of mean pore velocity. The function
Pe
f(Pe,b) = Pe + 4b2 + 2
(2.5.9)
where Pe = LV/D d , b = L/ü is a parameter describing the shape of channels
in the porous medium with L as their mean length and ü the characteristic
length of their cross sections. Since Pe depends on the molecular diffusion
coefficient D d , from Eqs. (2.5.1) and (2.5.8), we can find that the mechanical
dispersion coefficient, D;j' also depends on D d • This means that mechanical
dispersion coefficient and molecular diffusion coefficient are actually interrelated. In a hydrodynamic dispersion process, molecular diffusion has two
effects. One is the mean macroscopic effect of microscopic diffusion, which
exists even when the mean flow velocity is zero. The other is that diffusion
will cause the solute to transfer between flowlines, and hence, affect the
progress of mechanical dispersion. Normally, the second effect is relatively
weak and f(Pe, 15), as in Eq. (2.5.8), can be approximated as unity.
In a three-dimensional flow, the dispersivity tensor generally has 81 components. However, for an isotropie medium, only 36 of them are non-zero.
All of them are related to two constants, IX L and IX T • Scheidegger (1961) suggested the following expression:
(2.5.10)
where bij is the Kronecker delta, which is defined as
b .. = {O when i =F j
'J
1 when i = j,
(2.5.11)
and b mn , bim and so forth, have the same meanings as b ij • In Eq. (2.5.10), IX L is
called the longitudinal dispersivity of the isotropie medium and IXT the transversal dispersivity of the isotropie medium.
Substituting Eq. (2.5.10) into Eq. (2.5.8) and letting f(Pe, b) = 1, we then
have
(2.5.12)
37
expressions of dispersion coefficients, theoretical models have to be used.
Bear and Bachmet (1967) and Bear (1972) recommended the following expression for mechanical dispersion coefficients using their statistical geometrical model (Bear, 1972):
,
VmVn
~
Dij = IXijmn ----=- f(Pe, u),
V
(2.5.8)
where Einstein's summation convention is implied. IXijmn is a component of a
fourth rank tensor, called the dispersivity ofporous media, and has the dimension of length. V m and V n (m, n = 1,2,3) are components of mean pore velocity. The function
Pe
f(Pe,b) = Pe + 4b2 + 2
(2.5.9)
where Pe = LV/D d , b = L/ü is a parameter describing the shape of channels
in the porous medium with L as their mean length and ü the characteristic
length of their cross sections. Since Pe depends on the molecular diffusion
coefficient D d , from Eqs. (2.5.1) and (2.5.8), we can find that the mechanical
dispersion coefficient, D;j' also depends on D d • This means that mechanical
dispersion coefficient and molecular diffusion coefficient are actually interrelated. In a hydrodynamic dispersion process, molecular diffusion has two
effects. One is the mean macroscopic effect of microscopic diffusion, which
exists even when the mean flow velocity is zero. The other is that diffusion
will cause the solute to transfer between flowlines, and hence, affect the
progress of mechanical dispersion. Normally, the second effect is relatively
weak and f(Pe, 15), as in Eq. (2.5.8), can be approximated as unity.
In a three-dimensional flow, the dispersivity tensor generally has 81 components. However, for an isotropie medium, only 36 of them are non-zero.
All of them are related to two constants, IX L and IX T • Scheidegger (1961) suggested the following expression:
(2.5.10)
where bij is the Kronecker delta, which is defined as
b .. = {O when i =F j
'J
1 when i = j,
(2.5.11)
and b mn , bim and so forth, have the same meanings as b ij • In Eq. (2.5.10), IX L is
called the longitudinal dispersivity of the isotropie medium and IXT the transversal dispersivity of the isotropie medium.
Substituting Eq. (2.5.10) into Eq. (2.5.8) and letting f(Pe, b) = 1, we then
have
(2.5.12)