40
2. Hydrodynamic Dispersion in Porous Media
where r is the radius vector and h; is called the size Jactor. It can be proved
that in an orthogonal curvilinear co ordinate system the divergence of vector
A can be expressed as
.
1 [0
0
0
]
dlV A = - h
h h -:l(hzh3 A d + -:l(h l h3 A z ) + -:l(h l hzA 3 ) , (2.6.4)
I z 3 uU I
uU z
uU3
where Al' A z , and A 3 are the components of A in the curvilinear coordinates.
Furthermore, for any function ,p having continuous second-order partial
derivatives, the equation given below is true:
div(grad,p) = _1_[~(hzh3~) + ~(hlh3~) + ~(hlhz ~)]
h l h z h 3 oU I h l oU I
oU z h z oU z
OU3 h 3 oU 3 .
(2.6.5)
The general concepts of curvilinear coordinates and the derivations of Eqs.
(2.6.4) and (2.6.5) can be found in the field theory of advanced calculus.
Now, assurne that all the three axes in the orthogonal coordinate system
coincide with the principal axes of the dispersion coefficient tensor, e.g., axis
U I is of the same direction as the mean flow velo city, axes Uz and U 3 intersect
orthogonally with U I . For an isotropie porous medium, from Eq. (2.5.17) we
have
(2.6.6)
where VI = V and V z = V 3 = O. Combining Eqs. (2.6.4) and (2.6.5) with the
dispersion equation given in Eq. (2.4.24), we then have
oC
1 {o [h zh3
OC]
0 [h l h3
OC]
- = - - -
--(aL V + DdT)- + - --(aTV + DdT)ot
h l h z h 3 oU I h l
oU I
oU z h z
oU z
o [hlh z
OC]
0
}
+ - -(aTV + DdT)-:l - -:l(hzh3CV) .
oU 3 h 3
uU 3
uU I
(2.6.7)
Some examples are given below.
Cylindrical Coordinates (r, e, z)
The co ordinate transformation is given by Xl = r cos e, Xz = r sin e, X3 = z.
We then have the size factors h l = 1, hz = r, h3 = 1. By assuming that the
mean flow velocity has the same direction as radius r, the advection-dispersi on equation (2.6.7) becomes
OC 1 0 [
OC]
1 0 [
OC]
at = 2 or reaL V + Dd T)ar: + rZ oe (aT V + Dd T) oe
o [
OC] 1 0
+ OZ (aT V + Dd T) OZ - r or (rCV).
(2.6.8)