1. The Advection Dispersion Equation
By Michelle LeBaron
BAE 558
Spring 2007

2. What is the ADE? ADE – Advection Dispersion Equation
An equation used to describe solute transport through a porous media
Mechanisms:
Advection
Diffusion
Dispersion

3. Mechanisms Advection: the bulk movement of a solute through the soil
Diffusion: the movement of solutes caused by molecular movement that happens at the microscopic level
It causes solutes to move from areas of high concentration to areas of low concentration and is governed by Fick’s law.
Dispersion: a mixing that occurs because of the different velocities of neighboring flow paths.
This process occurs at many different levels and its affects increase as the scale increases.

4. Mechanisms Dispersion Continued
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5. Mechanisms Dispersion Continued
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6. Derivation
Application of the conservation of mass on a representative elementary volume (REV)
Analyze flux terms
Analyze sources and sinks term

7. Conservation of Mass Mass Balance on REV Where: S = surface area
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Where:
S = surface area
J = 3D vector flux
n = unit normal vector over S
Equation 1
Net volume leaving surface
Source and Sinks leaving surface
Change in mass of solute in volume over time

8. Conservation of Mass Equation 2
The gauss divergence theorem (equation 2) turns surface integrals into a volume integrals
Equation 2
By applying this to the flux term below from equation 1 we get equation 3
Flux Term
Equation 3

9. Conservation of Mass Equation 4
Bring everything over to one side to get equation 4
Equation 4
If the integral = 0 then everything inside the integral = 0 giving Equation 5 and rearranging terms to get equation 6
Equation 5
Equation 6

10. Flux Term, J
The vector flux term J is made of 3 components: advection (Jadv), diffusion (Jdiff), and dispersion (Jdisp)
Advective Transport: mass / time going through dA:

11. Flux Term, J Diffusion Transport: mass / time going through dA
Fick’s Law: The molecules are always moving and tend to move away from origin. This makes the diffusive flux proportional the concentration gradient of the solute in a direction normal to dA
Total Microscopic Flux:
Equation 7

12. Flux Term, J Dispersive Transport:
The dispersion component is due to variation in individual particles compared to the average velocity
. Depending on where they are in the flow path some will flow faster or slower than the average flow
The equation for the local velocity is the average velocity plus the deviation from the average velocity as can be seen in equation 8. A similar effect can be seen in equation 9 on the local concentration of a solute.
Equation 8
Equation 9

13. Flux Term, J
By substituting these effect of dispersion on the above term into equation 7 you get equation 10
Equation 10
Now we multiply the equation by and the fraction of the volume taking part in the flow and multiply the equation out to get the average flux with dispersion considered in Equation 11
Equation 11
Now note that the average deviation is zero to get Equation 12
Equation 12

14. Flux Term, J
This gives us the Jdisp:
Big assumption: It is not practical to track all the variations in concentration and velocity at every point at the macroscopic scale and we see that the variation increases with scale, so we assume it follow a “random walk” scheme that can now be modeled using Fick’s Law just like diffusion. This gives us the Jdisp term where D is a dispersion Coefficient and is a second rank tensor

15. Flux Term, J
Now add all of our flux terms to get the macroscopic flux found in equation 13
Finally by substituting J back into mass balance equation (equation 6) we get the ADE (equation 14)
Equation 6
Equation 14

16. Dispersion Coefficient
D is a tensor term and can be expanded as seen in equation below
By aligning D with the velocity you get the simplified equation shown below
By Taking the two transverse dispersions to be equal D you get an even more simplified equation shown below

17. Dispersion Coefficient
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18. References
Derivation of Advection/Dispersion Equation for Solute Transport in Saturated Soils
Selker, J. S., C.K. Keller, and J.T. McCord. Vadose Zone Processes. CRC Press LLC. Boca Raton Florida.1999.
Williams, Barbara. Solute Part 2 Lecture Notes.
Williams, Barbara. Dispersive Flux and Solution of the ADE.