1. Advection-Dispersion Equation (ADE)
Derivation of the
Advection-Dispersion Equation (ADE)
Assumptions
Equivalent porous medium (epm)
(i.e., a medium with connected pore space
or a densely fractured medium with a single
network of connected fractures)
Miscible flow
(i.e., solutes dissolve in water; DNAPL’s and
LNAPL’s require a different governing equation.
See p. 472, note 15.5, in Zheng and Bennett.)

2. 3. No density effects Density-dependent flow requires
a different governing equation. See
Zheng and Bennett, Chapter 15.
Figures from Freeze & Cherry (1979)

3. Advection-Dispersion Equation (ADE)
Derivation of the
Advection-Dispersion Equation (ADE)
Darcy’s law: h1 h1 h2 h2
q = Q/A
advective flux
fA = q c s s
f = F/A

4. How do we quantify the dispersive flux? h1 fA = advective flux = qc h2
f = fA + fD h2 s
How about
Fick’s law of diffusion?
where Dd is the effective
diffusion coefficient.

5. Dual Porosity
Domain
Figure from Freeze & Cherry (1979)
Fick’s law describes diffusion of ions on a molecular scale as ions diffuse from areas of higher to lower concentrations.

6. We need to introduce a “law” to describe
dispersion, to account for the deviation of
velocities from the average linear velocity
calculated by Darcy’s law.
Average linear velocity
True velocities

7. We will assume that dispersion follows
Fick’s law, or in other words, that dispersion
is “Fickian”. This is an important assumption;
it turns out that the Fickian assumption is not
strictly valid near the source of the contaminant.
porosity
where D is the dispersion coefficient.

8. Porosity () for example: q c = v  c
Mathematically, porosity functions as a kind of units conversion factor.
for example:
q c = v  c
Later we will define the dispersion coefficient
in terms of v and therefore we insert  now:

9. Case 1
Assume 1D flow qx s
and a line source

10. Case 1 qx porosity Advective flux Assume 1D flow Dispersive flux s
D is the dispersion coefficient. It includes
the effects of dispersion and diffusion. Dx is sometimes
written DL and called the longitudinal dispersion coefficient.

11. Assume 1D flow qx s
Case 2
and a point source

12. fA = qxc Advective flux Dispersive fluxes
Dx represents longitudinal dispersion (& diffusion);
Dy represents horizontal transverse dispersion (& diffusion);
Dz represents vertical transverse dispersion (& diffusion).

13. Continuous point source
Average
linear
velocity
Instantaneous point source
center of mass
Figure from Freeze & Cherry (1979)

14. Instantaneous Point Source Gaussian longitudinal dispersion transverse
Figure from Wang and Anderson (1982)

15. Derivation of the ADE for 1D uniform flow and 3D dispersion
(e.g., a point source in a uniform flow field)
vx = a constant vy = vz = 0
f = fA + fD
Mass Balance:
Flux out – Flux in = change in mass

16. Porosity () There are two types of porosity in transport problems:
total porosity and effective porosity.
Total porosity includes immobile pore water, which contains solute and therefore it should be accounted for when determining the total mass in the system.
Effective porosity accounts for water in interconnected pore space, which is flowing/mobile.
In practice, we assume that total porosity equals effective
porosity for purposes of deriving the advection-dispersion eqn.
See Zheng and Bennett, pp

17. Dx = xvx + Dd Dy = yvx + Dd Dz = zvx + Dd
Definition of the Dispersion Coefficient
in a 1D uniform flow field
Dx = xvx + Dd
Dy = yvx + Dd
Dz = zvx + Dd
vx = a constant
vy = vz = 0
where x y z are known as dispersivities. Dispersivity is essentially a “fudge factor” to account for the deviations of the true velocities from the average linear velocities calculated from Darcy’s law.
Rule of thumb: y = 0.1x ; z = 0.1y

18. ADE for 1D uniform flow and 3D dispersion
No sink/source term; no chemical reactions
Question: If there is no source term, how does the contaminant enter the system?

19. Simpler form of the ADE
Uniform 1D flow; longitudinal dispersion;
No sink/source term; no chemical reactions
Question: Is this equation valid for both point
and line sources?
There is a famous analytical solution to this form of the ADE with a continuous line source boundary condition. The solution is called the Ogata & Banks solution.

20. Effects of dispersion on the concentration profile
no dispersion
dispersion t1 t2 t3 t4
(Freeze & Cherry, 1979, Fig. 9.1)
(Zheng & Bennett, Fig. 3.11)

21. Effects of dispersion on the breakthrough curve

22. Instantaneous Point Source Gaussian
Figure from Wang and Anderson (1982)

23. Breakthrough
curve
long tail
Concentration
profile

24. Microscopic or local scale dispersion
Figure from Freeze & Cherry (1979)

25. Macroscopic Dispersion
(caused by the presence of heterogeneities)
Homogeneous aquifer
Heterogeneous
aquifers
Figure from Freeze & Cherry (1979)

26. Dispersivity () is a measure of the
heterogeneity present in the aquifer.
A very heterogeneous porous medium
has a higher dispersivity than a slightly
heterogeneous porous medium.

27. Dispersion in a 3D flow field

28. [K] = [R]-1 [K’] [R] global local z z’ x’ x Kxx Kxy Kxz Kyx Kyy Kyz x
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz
K’x
0 K’y 0
K’z
K =
[K] = [R]-1 [K’] [R]

30. Dispersion Coefficient (D)
D = D + Dd
D represents dispersion
Dd represents molecular diffusion
Dxx Dxy Dxz
Dyx Dyy Dyz
Dzx Dzy Dzz
D =
In general: D >> Dd

32. Dx = xvx + Dd Dy = yvx + Dd Dz = zvx + Dd
In a 3D flow field it is not possible to simplify the dispersion
tensor to three principal components. In a 3D flow field,
we must consider all 9 components of the dispersion tensor.
The definition of the dispersion coefficient is more complicated for 2D or 3D flow. See Zheng and Bennett, eqns
Dx = xvx + Dd
Dy = yvx + Dd
Dz = zvx + Dd
Recall, that for
1D uniform flow:

33. General form of the ADE:
Expands to 9 terms
Expands to 3 terms
(See eqn in Z&B)