longitudinal dispersion in nonuniform isotropic porous media

39
governed by an effective advection-diffusion equation for the mean flow
through the tube, with the dispersion along the tube given by the
results of Taylor's (1953) analysis. The mean velocity through a tube
is given by Poiseuille's law and the projection of the macroscopic
pressure gradient. The longitudinal component of the velocity along a
tube (denoted by u(t» is the sum of the longitudinal component of the
mean velocity plus a random component due to the effects of Taylor
dispersion. Averaging the Lagrangian correlation function over all
network configurations and integrating the time difference from
o to 00 , Saffman finds
DL 1 3
---=-+-
D 3 80
2 2
Pe + Pe
62 4
where d/a = pore length/pore radius
w = cos(8)
M = (3/2)Pe . w/D e
D 1 + (3/16)(Pe . w/6)2
e
M coth M - 1 dw
D M2
e
8 = direction of motion relative to mean
flow direction
The final integral is a function of Pee let number (V dIn) and 6 •
s
(2.14)
Equation (2.14) is plotted as a function of Pee let number in Figure 2.5
for 6 -+- 00 •
Following a similar procedure, Saffman calculates the lateral
dispersion coefficient
DT 1 1 Pe 2
9 2
-= - +-
Pe
62 - + D 3 80 8
a
.f w
1
2 (l
2 M coth M - 1
- w )
dl,)
D M2
e
(2.15)