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.

MechanismsDispersion Continued

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

MechanismsDispersion 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

Equation 1dV

t

CdVdS

S v v

r

)( nJ

Where:

S = surface area

J = 3D vector flux

n = unit normal vector over S

Net volume leaving surface

Source and Sinks leaving surface

Change in mass of solute in volume over time

Image from :http://www.webs1.uidaho.edu/age558/powerpoints/Solute_Part2.ppt

Conservation of Mass

The gauss divergence theorem (equation 2) turns surface integrals into a volume integrals

dVkdSkS

V )()( n

Equation 2

dVdSV

S )()( JnJ

By applying this to the flux term below from equation 1 we get equation 3

Flux Term

dVdt

dCdVdV

V

r

VV )( J

Equation 3

Conservation of Mass

Bring everything over to one side to get equation 4

0])[( dVdt

dC r

VJ

If the integral = 0 then everything inside the integral = 0 giving Equation 5 and rearranging terms to get equation 6

0)( dt

dC r

J

)( J

t

C r

Equation 4

Equation 5

Equation 6

Flux Term, J The vector flux term J is made of 3 components:

advection (Jadv), diffusion (Jdiff), and dispersion (Jdisp)

dAdAC advrl )()( nJnu

rladv CuJ

dispdiffadv JJJJ

Advective Transport: mass / time going through dA:

Flux Term, J

dAdACDdAC

D diffrl

rl )()( nJnn

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

rldiff CDJ

Total Microscopic Flux:

rl

rldiffadv CDC uJJJ

Equation 7

Flux Term, JDispersive 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.

uuu

rl

rl

rl CCC

Equation 8

Equation 9

Flux Term, JBy substituting these effect of dispersion on the above term into equation 7 you get equation 10

)())(( rl

rl

rl

rl CCDCC uuJ

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

Now note that the average deviation is zero to get Equation 12

))()()(( averlave

rlave

rl CDCC uuJ

Equation 12

))()()()()()(( averlave

rlave

rlave

rlave

rlave

rl CDCDCCCC uuuuJ

Equation 11

Flux Term, J

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

averldisp C )( uJ

rdisp C DJ

This gives us the Jdisp:

Flux Term, J

rrdispdiffadv CDC )( DuJJJJ

Now add all of our flux terms to get the macroscopic flux found in equation 13

))(( rr

r

CDCt

CDu

Finally by substituting J back into mass balance equation (equation 6) we get the ADE (equation 14)

)( J

t

C r

Equation 6

Equation 14

Dispersion Coefficient

zz

yy

xx

D

D

D

00

00

00

D

zzyzxz

zyyyxy

zxyxxx

DDD

DDD

DDD

D

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

T

L

D

D

0

0D

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