The Advection Dispersion Equation By Michelle LeBaron BAE 558 Spring 2007

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

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.

Mechanisms Dispersion Continued Images From:http://age-web.age.uiuc.edu/classes/age357/html/Advective%20Dispersive%20Equation.pdf

Mechanisms Dispersion Continued Images From:http://age-web.age.uiuc.edu/classes/age357/html/Advective%20Dispersive%20Equation.pdf

Derivation Application of the conservation of mass on a representative elementary volume (REV) Analyze flux terms Analyze sources and sinks term

Conservation of Mass Mass Balance on REV Where: S = surface area Image from :http://www.webs1.uidaho.edu/age558/powerpoints/Solute_Part2.ppt 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

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

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

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:

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

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

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

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

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

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

Dispersion Coefficient Image From:http://www.webs1.uidaho.edu/age558/powerpoints/Solute_Mon_Apr_15.ppt

References Derivation of Advection/Dispersion Equation for Solute Transport in Saturated Soils http://age-web.age.uiuc.edu/classes/age357/html/Advective%20Dispersive%20Equation.pdf 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. http://www.webs1.uidaho.edu/age558/powerpoints/Solute_Part2.ppt Williams, Barbara. Dispersive Flux and Solution of the ADE. http://www.webs1.uidaho.edu/age558/powerpoints/Solute_Mon_Apr_15.ppt