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2. Hydrodynamic Dispersion in Porous Media
In the case ofthree-dimensional flow, Eq. (2.5.12) determines the nine components of the mechanical dispersion coefficient tensor,
theyare
D~l = [aT(Vl + vl) + aL vn/v
D~z = D;l = (aL - aT)Vl vz/v
D~3 = D;l = (aL - aT)Vl v3/v
D;z = [aT(V/ + vl) + aL vn/v
D;3 = D;z = (aL - aT)Vz v3/v
D;3 = [aT(V/ + vl) + aLvn/v
(2.5.13)
(2.5.14)
Ifaxis Xl of the Cartesian coordinates coincides with the me an flow direction, and both axes X z and X 3 are perpendicular to it, Eq. (2.5.14) can be
simplified to
D~ 1 = aL V; D;z = D;3 = aT V; D;j = 0 (i '" j),
and thus, we have
(2.5.15)
(2.5.16)
The axes defined in this way are called the principal axes of dispersion.
If a flow fieId is not unidirectional, the principal axes of dispersion will
change with the velocity fieId. The dispersion coefficient is structurally different from that of permeability, because even in an isotropie medium it is still
a tensor, and its principal axes are determined by the mean flow direction,
rather than the medium.
The coefficient of total dispersion is the sum of the mechanical dispersion
coefficient, D', and the molecular diffusion coefficient in porous media, D" =
D d T, where D d is the molecular diffusion coefficient in solution, and T is the
tortuosity ofthe porous medium. For an isotropie porous medium, it reduces
to a seal ar T, and 0 < T< 1. Thus, the coefficient of total dispersion can be
expressed as
(2.5.17)