2.5. Coefficients of Hydrodynamic Dispersion
33
According to mass conservation, the sum of the above three factors should
be equal to the solute increase AM in liquid phase of (R), i.e.,
(2.4.25)
From the physical meanings of the terms mentioned above, we have
M 1 = - r OCV . n dS;
(2.4.26)
J(S)
M 2 = - r OJ*.ndS;
JS)
M 3 = r OidR,
JR)
(2.4.27)
(2.4.28)
where CV is the advection flux, J* the hydrodynamic dispersion flux, 0 the
volumetrie fraction of the liquid phase, and oi gives the solute production
rate in (R), that is, the mass produced in (R) per unit volume and per unit
time. Note that the mass of solute per unit volume in a porous medium is OC,
therefore
AM = i o(OC)dR.
Ll
(2.4.29)
(R)
ot
Substituting Eqs. (2.4.26) through (2.4.29) into Eq. (2.4.25), and using the
linear dispersion law
J* = - D gradC,
(2.4.30)
we then have
i
o(OC)
i - i -- i-a - dR = OD gradC . n dS -
OCV· n dS +
01 dR.
(R)
t
(S)
(S)
(R)
(2.4.31)
This is the integral form of the hydrodynamic dispersion equation with C
unknown. It is evident from the derivation above that the equation is also a
combination ofthe mass conservation equation (2.4.25) and the linear dispersion law equation (2.4.30). Similar to the groundwater flow equations, the
transfer between integral and differential forms of the hydrodynamic dispersion equations can be done by using Green's formula.
2.5 Coefficients of Hydrodynamic Dispersion
2.5.1 Coefficients of Longitudinal Dispersion and
Transverse Dispersion
The coefficient of hydrodynamic dispersion is a tensor, but its tensor characteristics will not be easily recognized if the study is restricted to one-dimensional dispersion phenomena. That is why in the early studies the dispersion